If I may, you are asking a couple questions here that I believe are a bit in conflict.
Concerning your last question, "Are there anything people should keep in mind to avoid hyperthyroidism while eating Paleo?" I can answer quite certainly, if philosophically, no. The tenet of the Paleo way of eating is that it alone is the proper, no compromises manner of consumption for which humans are evolutionarily suited. If you accept that tenet then you cannot fear Paleo could potentially lead to a chronic condition like hyperthyroid.
BTW, that tenet is an important point. From it follows the important distinction between simple low carb dieting and true Paleo way of eating. A simple low carb diet will allow someone to lose weight and improve many of their health markers. However, low carb-ing on KFC, McD, and whatever other hydrogenated seed oils we are bound to get from Big-Ag once low carb has grabbed the mainstream attention is not Paleo. Paleo encompasses the right fats, the right vitamins, and the right minerals in proper portion from natural, whole foods.
Now, the topic question was "Can Paleo exacerbate hyperthyroidism?" That is an altogether different question than "can Paleo lead to hyperthyroidism". In fact, I would rephrase it with emphasis as: "Are there aspects of a Paleo style of eating which may exacerbate a pathological cause of hyperthyroidism?"
I think the emphasis are important. Without those emphasis one may jump to fearing that Paleo may lead to hyperthyroid (or any chronic condition). As to the potentially exacerbating aspects, as concerns hyperthyroid I don't have an answer. I will say there are other conditions for which aspects of Paleo can have a negative effect. For instance, if you are already iodine deficient or very close to being so, the lack of iodine laced salt that comes with the processed food in the SAD can be problematic. One must mitigate that aspect of Paleo (ie. no industrially processed and manufactured food) by eating fish, seaweed or supplementing. Another condition is PKU. Meat causes these people harm. They must instead consume dairy for protein and non-grain based complex carbohydrates. Yet another aspect of Paleo is eating nutrient dense food and lots of animal fat. One must mitigate those aspects with moderation and exercise. Beef liver appears to be nature's multi-vitamin, however, like commercial multi-vitamins you can overdose on them, so moderation is key. Animal fats from commercial commodity producers can be high in omega-6 fats. One must mitigate again by moderation, preferring ruminants over pork and chicken, and grass-fed over feed lot corn fed, and perhaps supplementing omega-3 capsules.
Finally, this may be more of a philosophical answer than you were expecting (you can thank a Friday home with the kids). If so I'll hold no ill will for missing the bounty (as much as I would like to win it!). I do think it is an important answer though.
