(Warning, this is quite long. Conclusions in bold at the bottom.)

First, I will argue that when T3 becomes lower in a low carbohydrate setting, it does so with the function of sparing muscle, because the diet has a deficit of calories or protein, which creates a catabolic state. If the T3 were not lower, or if it were supplemented by a well meaning person who interpreted this lowering as simply hypothyroid, then lean mass would be lost.

The fact is that in weight loss, starvation, or protein deficiency conditions, lowered T3 is thought to be a functional response that protects against muscle loss, effectively ensuring the favourable body composition effects of a low carb diet as compared to a higher carb diet. That a low carb diet has this advantage is not particularly controversial, and T3 is thought to be the mechanism. Here are a couple of references.

Relationship between the changes in serum thyroid hormone levels and protein status during prolonged protein supplemented caloric deprivation.

The relationship between the changes in serum thyroid hormone levels and nitrogen economy during caloric deprivation were investigated in ten obese men during a 40 d, 400 kcal protein-supplemented weight-reducing diet. This regimen induced increases in the serum levels of total T4, free T4 and total rT3, and decreases of total T3, while serum TSH remained unchanged. There were progressive decreases in total body weight and urinary losses of total nitrogen and 3-methylhistidine, with the early negative nitrogen balance gradually returning towards basal values during the 40 days. Subjects with the largest weight loss had the most increase in the serum levels of total T4 and free T4 index and the greatest decrease in T3. The magnitude of the increase of the nitrogen balance from its nadir was correlated with the extent of the reduction of T3 and increase of T3 uptake ratio and free T4 levels. The decrease in the urinary excretion of 3-methylhistidine correlated with the increase in free T4 and rT3 levels. Nadir serum transferrin values were directly related to peak rT3 values, and the lowest albumin concentrations occurred in subjects with the highest total T4 and free T4 index values. Further, the maximum changes in the serum thyroid hormone levels preceded those of the nutritional parameters. These relationships suggest that: (1) increases in serum rT3 and free T4 and reductions in T3 concentrations during protein supplemented weight reduction may facilitate conservation of visceral protein and reduce muscle protein turnover; and (2) the variation in the magnitude of these changes may account for the heterogeneity of nitrogen economy.

Variability in body protein loss during protracted, severe caloric restriction: role of triiodothyronine and other possible determinants

From the abstract:

Although the rate of fat loss was relatively constant throughout the study, wide interindividual variations in cumulative protein (nitrogen) deficit were observed. Total nitrogen losses per subject ranged from 90.5 to 278.7 g. Cumulative nitrogen loss during the first 16 days tended to correlate negatively with initial mean fat cell size and positively with initial lean body mass. Most notable was the strong negative correlation between the size of the decrease in serum triiodothyronine over the 64-day study and the magnitude of the concurrent cumulative N deficit. During severe caloric restriction, one's ability to decrease circulating serum triiodothyronine levels may be critical to achievement of an adaptational decrease in body protein loss.

Moreover, supplementing T3 in these conditions causes muscle loss, and is not desirable.

Metabolic responses in grossly obese subjects treated with a very-low-calorie diet with and without triiodothyronine treatment

From the abstract:

Thus, 74 per cent of the extra weight loss in the T3 treated group could be accounted for by loss of fat free tissue... Our results warrant the conclusion that there appears to be no place for T3 as an adjunct to dieting, as it enhances mostly body protein loss and only to a small extent loss of body fat.

In light of this, when we look at the four studies above that refer to very low calorie diets, we can see what is happening more clearly. Let's look at what they say.

Effect of caloric restriction and dietary composition on serum T3 and reverse T3 in man I'm including the whole abstract, because it is very explicit and unambiguous, which is fortunate, because I do not have access to its full text.

To evaluate the effect of caloric restriction and dietary composition on circulating T3 and rT3, obese subjects were studied after 7–18 days of total fasting and while on randomized hypocaloric diets (800 kcal) in which carbohydrate content was varied to provide from 0 to 100% calories. As anticipated, total fasting resulted in a 53% reduction in serum T3 in association with a reciprocal 58% increase in rT3. Subjects receiving the no-carbohydrate hypocaloric diets for two weeks demonstrated a similar 47% decline in serum T3 but there was no significant change in rT3 with time. In contrast, the same subjects receiving isocaloric diets containing at least 50 g of carbohydrate showed no significant changes in either T3 or rT3 concentration. The decline in serum T3 during the no-carbohydrate diet correlated significantly with blood glucose and ketones but there was no correlation with insulin or glucagon. We conclude that dietary carbohydrate is an important regulatory factor in T3 production in man. In contrast, rT3, concentration is not significantly affected by changes in dietary carbohydrate. Our data suggest that the rise in serum rT3 during starvation may be related to more severe caloric restriction than that caused by the 800 kcal diet.

So at least in a very low calorie situation, T3 becomes low only when the diet is sufficiently low in carbohydrate to be ketogenic, and its level correlates with ketogenesis. We are not told whether any of the diets were protein sufficient, but in this case it doesn't matter. The very low calories make it catabolic, and only when carbohydrate is at ketogenically low levels does the protein sparing effect occur.

This one is similar.

The effect of varying carbohydrate content of a very-low-caloric diet on resting metabolic rate and thyroid hormones

Twelve obese women were studied to determine the effects of the combination of an aerobic exercise program with either a high carbohydrate (HC) very-low-caloric diet (VLCD) or a low carbohydrate (LC) VLCD diet on resting metabolic rate (RMR), serum thyroxine (T4), 3,5,3′-triiodothyronine (T3), and 3,5′,3′-triiodothyronine (rT3). The response of these parameters was also examined when subjects switched from the VLCD to a mixed hypocaloric diet. Following a maintenance period, subjects consumed one of the two VLCDs for 28 days. In addition, all subjects participated in thrice weekly submaximal exercise sessions at 60% of maximal aerobic capacity. Following VLCD treatments, participants consumed a 1,000 kcal mixed diet while continuing the exercise program for one week. Measurements of RMR, T4, T3, and rT3 were made weekly. Weight decreased significantly more for LC than HC. Serum T4 was not signficantly affected during the VLCD. Although serum T3 decreased during the VLCD for both groups, the decrease occurred faster and to a greater magnitude in LC (34.6% mean decrease) than HC (17.9% mean decrease). Serum rT3 increased similarly for each treatment by the first week of the VLCD. Serum T3 and rT3 of both groups returned to baseline concentrations following one week of the 1,000 kcal diet. Both groups exhibited similar progressive decreases in RMR during treatment (12.4% for LC and 20.8% for HC), but values were not significantly lower than baseline until week 3 of the VLCD. Thus, although dietary carbohydrate content had an influence on the magnitude of fall in serum T3, RMR declined similarly for both dietary treatments.

The difference here was that even on the higher carb diet, there was some T3 lowering.

Although they note that T3 and rT3 values both returned to baseline, the chart in the paper does not seem to show that. For T3, the LC group started at 122+-10, went down to 79+-8 during restriction, and returned to 114+-12 within a week post. The HC group started at 118+-23, went down to 87+-10, and only returned to 98+-3. For rT3, the LC group started at 20.9+-1.1, went up to 25.7+-1.8, and returned to 19.6+-1.9. The HC group started at 22.2+-1.0, went up to 26.6+-2.2, and returned to 23.4+-1.8. These differences were not statistically significant.

Incidentally, they note in the paper that resting metabolic rate did not return to normal for either group after one week, which could have regain implications. I think this is not uncommon with very low calorie diets.

Here is a 3rd study of low calorie diets of various compositions:

Effect of dietary carbohydrates during hypocaloric treatment of obesity on peripheral thyroid hormone metabolism.

The effect of different hypocaloric carbohydrate (CHO) intakes was evaluated in 8 groups of obese patients in order to assess the role of the CHO and the other dietary sources in modulating the peripheral thyroid hormone metabolism. These changes were independent of those of bw. Serum T3 concentrations appear to be more easily affected than those of reverse T3 by dietary manipulation and CHO content of the diet. A fall in T3 levels during the entire period of study with respect to the basal levels occurred only when the CHO of the diet was 120 g/day or less, independent of caloric intake (360, 645 or 1200 calories). Moreover, reverse T3 concentrations were found increased during the entire period of study when total CHO were very low (40 to 50 g/day) while they demonstrated only a transient increase when CHO were at least 105 g/day (with 645 or more total calories). Indeed, our data indicate that a threshold may exist in dietary CHO, independent of caloric intake, below which modifications occur in thyroid hormone concentrations. From these results it appears that the CHO content of the diet is more important than non-CHO sources in modulating peripheral thyroid hormone metabolism and that the influence of total calories is perhaps as pronounced as that of CHO when a "permissive" amount of CHO is ingested.

Again, all of the diets were hypocaloric, and there was a threshold of carboydrate intake, below which T3 lowered.

The fourth is yet another low calorie diet varying composition.

Effect of dietary composition on fasting-induced changes in serum thyroid hormones and thyrotropin

To assess the effect of starvation and refeeding on serum thyroid hormones and thyrotropin (TSH) concentrations, 45 obese subjects were studied after 4 days of fasting and after refeeding with diets of varying composition. All subjects showed an increase in both serum total and free thyroxine (T4), and a decrease in serum total and free triiodothyronine (T3) following fasting. These changes were more striking in men then in women. The serum T3 declined during fasting even when the subjects were given oral L-T4, but not when given oral L-T3. After fasting, the serum reverse T3 (rT3) rose, the serum TSH declined, and the TSH response to thyrotropin-releasing hormone (TRH) was blunted. Refeeding with either a mixed diet (n = 22) or a carbohydrate diet (n = 8) caused the fasting-induced changes in serum T3, T4, rT3, and TSH to return to control values. In contrast, refeeding with protein (n = 6) did not cause an increase in serum T3 or in serum TSH of fasted subjects, while it did cause a decline in serum rT3 toward basal value.

The present data suggest that: (1) dietary carbohydrate is an important factor in reversing the fall in serum T3 caused by fasting; (2) production of rT3 is not as dependent on carbohydrate as that of T3; (3) men show more significant changes in serum thyroid hormone concentrations during fasting than women do, and (4) absorption of T3 is not altered during fasting.

Note that in this case, "refeeding" was with an 800 calorie diet.

Three of the studies were intended as evidence that high fat itself can have this effect. Unfortunately, it is not clear from these studies whether the results are more a reflection of high fat or low protein. Since we already know that low T3 is protein sparing, the latter is likely to be the explanation. Here are the studies.

Effect of low-carbohydrate diets high in either fat or protein on thyroid function, plasma insulin, glucose, and triglycerides in healthy young adults.

A low-carbohydrate diet, frequently used for treatment of reactive hypoglycemia, hypertriglyceridemia, and obesity may affect thyroid function. We studied the effects of replacing the deleted carbohydrate with either fat or protein in seven healthy young adults. Subjects were randomly assigned to receive seven days of each of two isocaloric liquid-formula, low-carbohydrate diets consecutively. One diet was high in polyunsaturated fat (HF), with 10%, 55%, and 35% of total calories derived from protein, fat, and carbohydrate, respectively. The other was high in protein (HP) with 35%, 30%, and 35% of total calories derived from protein, fat, and carbohydrate. Fasting blood samples were obtained at baseline and on day 8 of each diet. A meal tolerance test representative of each diet was given on day 7. The triiodothyronine (T3) declined more (P less than .05) following the HF diet than the HP diet (baseline 198 micrograms/dl, HP 138, HF 113). Thyroxine (T4) and reverse T3 (rT3) did not change significantly. Thyroid-stimulating hormone (TSH) declined equally after both diets. The insulin level was significantly higher 30 minutes after the HP meal (148 microU/ml) than after the HF meal (90 microU/ml). The two-hour glucose level for the HP meal was less, 85 mg/dl, than after the HF meal (103 mg/dl). Serum triglycerides decreased more after the HF diet (HF 52 mg/dl, HP 67 mg/dl). Apparent benefits of replacing carbohydrate with polyunsaturated fat rather than protein are less insulin response and less postpeak decrease in blood glucose and lower triglycerides. The significance of the lower T3 level is unknown.

This one appears to be comparing high protein to high fat, and showing a lower T3 in the high fat condition than the high protein condition. Although it does not explicitly say so, it would seem highly likely, based on the wording of the abstract, that these were both calorie restricted diets, and therefore we are in the condition I've already discussed. Again the more protein, the less this effect was seen, which is consistent with the protein sparing function of low T3.

Metabolic differences in response to a high-fat vs. a high-carbohydrate diet

Energy expenditure was measured in a group of 7 subjects who received two isocaloric isonitrogenous diets for a period of 9-21 days with a 4-10-day break between diets. Diet 1 was a high-fat diet ( 83.5 +/- 3.6% of total energy). Diet 2 was a high carbohydrate diet ( 83.1 +/- 3.7% of total energy). Resting and postprandial resting metabolic rate were measured by open circuit indirect calorimetry 2-4 times during each metabolic period. Total energy expenditure (TEE) was measured by the doubly labeled water method over an 8-13-day period. The respiratory quotient was measured 2-4 hours after a meal during each metabolic period for the calculation of total energy expenditure by the doubly labeled water method. Levels of total T3 (TT3), T3 uptake, free thyroid index and T4 were measured at the end of each metabolic period. No significant changes in resting metabolic rate (RMR) were apparent on the two diets (1567 +/- 426 kcal/d high-fat diet and 1503 +/- 412 kcal/d high-carbohydrate diet n=7, p<0.15). Total energy expenditure measured in 5 subjects was significantly higher during the high-carbohydrate phase of the diet (2443 +/- 422 vs. 2078 +/- 482 kcal/d p<0.05). Activity estimated from TEE/RMR was greater on the high-carbohydrate diet but only approached statistical significance (p<0.06). Total T3 was significantly lower and free thyroid index and T3 uptake were significantly higher at the end of the high fat diet in comparison to the high-carbohydrate diet. These data suggest that individual tolerance to a high-fat diet varies considerably and may significantly lower TEE by changing levels of physical activity. The explanation for changes in thyroid hor. mone levels independent of changes in metabolic rate remains unclear.

In this one, it also does not state explicitly that it is a calorie reduced diet, but at 15xx calories it may have been. In this case, both were approximately equally low in protein (probably significantly less than 60g, which almost everyone would agree is deficient), but one of them was high in carbohydrate, which, as we would expect, nullified the protective T3 lowering effect.

The role of dietary fat in peripheral thyroid hormone metabolism

Short term changes in serum 3,3′,5-triiodothyronine (T3) and 3,3′,5′-triiodothyronine (reverse T3, rT3) were studied in four healthy nonobese male subjects under varying but isocaloric and weight maintaining conditions. The four 1500 kcal diets tested during 72 hr, consisted of: 1, 100% fat; II, 50% fat, 50% protein; III, 50% fat, 50% carbohydrate (CHO), and IV, a mixed control diet. The decrease of T3 (50%) and increase of rT3 (123%) in the all-fat diet equalled changes noted in total starvation. In diet III (750 kcal fat, 750 kcal CHO) serum T3 decreased 24% (NS) and serum rT3 rose significantly 34% (p < 0.01). This change occurred in spite of the 750 kcal CHO. This amount of CHO by itself does not introduce changes in thyroid hormone levels and completely restores in refeeding models the alterations of T3 and rT3 after total starvation. The conclusion is drawn that under isocaloric conditions in man fat in high concentration itself may play an active role in inducing changes in peripheral thyroid hormone metabolism. This is finally a study that is explicitly a maintenance diet. It says mostly what we would expect. The 100% fat diet lowered T3. In the half protein, and "mixed" conditions, no lowering was reported. It was a bit surprising, and contrary to some previous findings, that in the half carb, half fat diet, this high a carbohydrate level would still allow lower T3. The authors suggest that this is evidence that high fat alone is responsible. I would suggest that perhaps it was because of the zero protein condition. In the body of the paper, they acknowledge they are speculating. Nonetheless, it is less important to know with certainty for our purposes, because we are trying to decide whether the low T3 seen in low carbohydrate conditions is problematic, not whether an additional observation of low T3 even with higher carbohydrate is.

The last study that I was asked to look at is more challenging to interpret, because it claims to find both low T3 and protein catabolism in their observations. Since low T3 putatively protects muscle catabolism, this observation would be problematic.

Isocaloric carbohydrate deprivation induces protein catabolism despite a low T3-syndrome in healthy men

Dietary carbohydrate content is a major factor determining endocrine and metabolic regulation. The aim of this study was to evaluate the relation between thyroid hormone levels and metabolic parameters during eucaloric carbohydrate deprivation.

We measured thyroid hormone levels, resting energy expenditure (by indirect calorimetry) and urinary nitrogen excretion in six healthy males after 11 days of three isocaloric diets containing 15% of energy equivalents as protein and 85%, 44% and 2% as carbohydrates.

In contrast to the high and intermediate carbohydrate diets, carbohydrate deprivation decreased plasma T3 values (1·78 ± 0·09 and 1·71 ± 0·07 vs. 1·33 ± 0·05 nmol/l, respectively, P < 0·01), whereas reverse T3, T3 uptake and free T4 levels increased simultaneously compared to the other two diets. TSH values were not different among the three diets. Although dietary carbohydrate content did not influence resting energy expenditure, carbohydrate deprivation increased urinary nitrogen excretion (10·91 ± 0·67 and 12·79 ± 1·14 vs. 15·89 ± 1·10 g/24 h, respectively, P = 0·03).

Eucaloric carbohydrate deprivation increases protein catabolism despite decreased plasma T3 levels. Because it has previously been shown that starvation decreases plasma T3 levels, resting energy expenditure and nitrogen excretion, these discordant endocrine and metabolic changes following carbohydrate deprivation indicate that the effects of starvation on endocrine and metabolic regulation are not merely the result of carbohydrate deprivation.

My analysis of this study, is that the protein levels are simply not high enough (although they state that they are using "normal" protein levels). If you administer a diet with low carbohydrates and insufficient protein, it is not reasonable to expect that low T3 will completely prevent muscle catabolism. The probelm is that they did not account for gluconeogenesis. Even if the protein levels were adequate in a non-ketogenic setting, it would be absurd to expect protein needs not to increase in the ketogenic one. So it is misleading to compare a ketogenic diet with a non-ketogenic diet given the same amount of protein. I consider this a major flaw in their reasoning.

A second problem was that the adaptation period was minimal.

Here is the result that is found when adequate protein is given and adequate time on a ketogenic diet with maintenance calories. The human metabolic response to chronic ketosis without caloric restriction: Physical and biochemical adaptation

To study the metabolic effects of ketosis without weight loss, nine lean men were fed a eucaloric balanced diet (EBD) for one week providing 35–50 kcal/kg/d, 1.75 g of protein per kilogram per day and the remaining kilocalories as two-thirds carbohydrate (CHO) and one-third fat. This was followed by four weeks of a eucaloric ketogenic diet (EKD)—isocaloric and isonitrogenous with the EBD but providing less than 20 g CHO daily. Both diets were appropriately supplemented with minerals and vitamins. Weight and whole-body potassium estimated by potassium-40 counting (40K) did not vary significantly during the five-week study. Nitrogen balance (N-Bal) was regained after one week of the EKD... These findings indicate that the ketotic state induced by the EKD was well tolerated in lean subjects; nitrogen balance was regained after brief adaptation, serum lipids were not pathologically elevated, and blood glucose oxidation at rest was measurably reduced while the subjects remained euglycemic.

My conclusion based on all of this is that the lowered T3 that is seen in a ketogenic diet when calories or protein are restricted serves an important function -- one that would be detrimental to undo. Other parameters indicating hypothyroid aren't experienced, and levels rise when the diet is stopped. Therefore, I do not agree with the assessment that ketogenic diets reduce thryoid function.

A second question might be, whether or not there are other, not so beneficial effects of lowered T3 that we ought to be wary of, when deciding whether to use a ketogenic diet for weight loss. While we can't know the answer to that with certainty, I have a couple of reasons for thinking not.

First, the T3 lowering appears to occur without negatively affecting other parts of the thyroid system. It is not equivalent to becoming hypothyroid. In fact, the phenomenon is sometimes called "euthyroid sick syndrome", the "euthyroid" meaning that otherwise, the thyroid is well. Hypothyroid can sometimes be associated with increased risk for cardiovascular disease, but this is only the case when it is accompanied by other CVD risk factors. In other words, when controlled for other risk factors the effect disappears.

Here is a review study on that topic.

Cardiovascular morbidity and mortality in thyroid dysfunction

Consistently, good evidence exists for an increased cardiovascular morbidity in overt hyperthyroidism... The cardiovascular risk profile of overt hypothyroidism is characterized mainly by risk factors of atherosclerosis such as hypercholesterolemia and hypertension... the evidence for similarly increased cardiovascular morbidity and mortality rates in subclinical hyperthyroidism and hypothyroidism is inconclusive, and the evidence is non-existent for overt hypothyroidism.

Since a ketogenic diet has an overwhelmingly positive effect on blood lipid profiles with respect to cardiovascular risk, it would not seem to be a concern.

Finally, low T3 is actually associated with longevity. For example:

Evaluation of neuroendocrine status in longevity

Our data revealed several differences in the neuroendocrine and metabolic status of centenarians, compared with other age groups, including the lowest serum concentrations of leptin, insulin and T3, and the highest values for prolactin. We failed to find any significant differences in TSH and cortisol levels.

This does not appear to be simply an effect of old age, but rather a genetic factor:

Low Serum Free Triiodothyronine Levels Mark Familial Longevity: The Leiden Longevity Study

Compared with their partners, the group of offspring of nonagenarian siblings show a lower thyroidal sensitivity to thyrotropin. These findings suggest that the favorable role of low thyroid hormone metabolism on health and longevity in model organism is applicable to humans as well.

As they mention, this effect is already believed to be part of the mechanism in animal longevity studies. For example:

Effect of Long-Term Calorie Restriction with Adequate Protein and Micronutrients on Thyroid Hormones

Caloric restriction (CR) retards aging in mammals. It has been hypothesized that a reduction in T3 hormone may increase life span by conserving energy and reducing free-radical production... Long-term CR with adequate protein and micronutrient intake in lean and weight-stable healthy humans is associated with a sustained reduction in serum T3 concentration, similar to that found in CR rodents and monkeys. This effect is likely due to CR itself, rather than to a decrease in body fat mass, and could be involved in slowing the rate of aging.

In conclusion, no, I do not consider it dishonest to minimize the observation that low carbohydrate diets can lower T3. Such lowering is protective of lean mass, and does not appear to be harmful or indicative of hypothyroid. It is completely reversible, which indicates that it is not causing an impairment of function, but rather is part of normal thyroid function. Finally, it may even be beneficial.