You can't exist without any sugar in the blood. You will always have some sugar in your system because some of your tissues have no mitochondria in the cells, or not enough to get by on fatty acid metabolism, so they need glucose instead. A few examples I can think of off the top of my head are mature red blood cells (no nuclei), certain types of nerve cells and certain cells in the testes, for men.
Now before you go "oh well then sugar must be necessary!", yes, in very small amounts. I believe Dr. Eades said that a healthy level of fasting glucose is equivalent to about a teaspoon of sugar for your entire body. And happily, you don't have to eat carbohydrate of any sort to produce it. Your body can make it from protein or glycerol. The definition of "essential nutrient" is "a nutrient your body cannot make but that you need to live." This is why zero-carb Paleo eaters aren't dropping dead in droves.
(Carb foods have their place, most notably for micronutrient intake, but you can live without them if you eat organ meats.)
Now then, what you're asking is about excess glucose. Your body can tolerate its own fasting level without clearing it out; the insulin release is for dealing with any glucose above that amount. But let's say you're not eating any glucose-producing foods.
Insulin is still necessary for dealing with amino acids, however. Any time you eat protein, your insulin will be elevated 'cause you've got to put those amino acids into your muscles. (Not being able to do this is why type 1 diabetics eventually die without insulin injections--diet helps the sugar problem, but they still waste away.) So any time you eat protein, your insulin will still be elevated in response. Actually it elevates in response to smelling food, too.
So it's normal for your body to deal with varying levels of insulin in the blood. That's OK, as long as insulin is not constantly elevated. What messes people up who have hyperinsulinism is that it's nearly always elevated, with deleterious effects on blood vessels and tumor growth and hormone imbalance and the like.
You don't want to inject the stuff though, not if your pancreas is already working properly. Best to let your body produce it as needed. What would happen if you injected it without a clear need is your blood sugar would drop to a dangerously low level, potentially killing you.
Insulin resistance in the absence of sugar may or may not be a problem. It depends on which tissues are insulin-resistant. As long as your muscles can still get their aminos and the cells that need glucose can still get glucose, you should be OK. Some degree of resistance is to be expected in certain medical situations (i.e., pregnancy causes a certain level of insulin resistance, which is why pregnant women are tested for gestational diabetes). If you're not dumping in a bunch of sugar on top of insulin resistance, you should be OK. What kills type 2 diabetics is the excess sugar, far more than the insulin resistance. It's possible to be type 2 but have normal blood sugar if you're eating sensibly (i.e., low-carbing, not following ADA advice!).