Does wearing a coat require your body to produce less heat? Question: If you’re wearing coat on a cold day, does this require your body to produce less heat to keep you warm? How can we model this energy exchange?
This question is inspired by the idea that you can supposedly burn more calories when you’re cold. (Article on the subject, question on the subject.)
This sounds simple, but my confusion arises from the fact that coats are imperfect insulators. If you're in Antarctica in a casual coat, I imagine it wouldn’t be much better than being naked. I’ve also heard anecdotally that you have to eat more when living in a cold region of the country, which sounds plausible.
Over a long period of time, how much of a difference does a coat make? Does it just slow down the loss, still requiring your body to create more heat?
 A: Yes.
Your body, along with all the waste heat from digestion, thinking, growing etc has to produce heat to keep your core temperature in a very small range around the normal 37c/98f.
If your surroundings are cold enough that the heat loss is too much to  maintain this temperature your body will try to do more work, your muscles will fire = you will shiver.
If you are better insulated, or you are in a warmer environment, then there is less heat loss and your body has to do less extra work to maintain your core temperature.
This assumes you are a mammal - if you are a reptile, fish or amphibian you will lower your core temperature, stop digesting, growing, thinking etc until you warm up again.
These US military rations suggest that you need 50% more calories in a cold environment, presumably even with cold weather clothing.
A: Think of the energy conservation law. Of you want to maintain a certain body temperature, then your body must be in thermodynamic equilibrium. All energy input every second must equal all energy output energy second - we could say: the rate of heating must equal the rate of cooling.
Your body has internal heating, from the nutrition you digest. 
Cooling happens from radiation and convection constantly. The coat is an insulator, that prevents both. Your cooling rate is now lower.
Now the heating rate is larger than that of cooling. You therefore heat up. Your temperature rises. There are now two options:


*

*Your body must adjust - that is, stop producing heat. Martin Beckett's answer gives some good points on different body mechanisms for adjustments in different life forms. Or

*you accept the change. Your temperature will continue to rise until the cooling rate equals the heating rate again. Since more energy is lost when the temperature is higher (your body heat will eventually heat up and through the coat), this equilibrium will happen at some point. Your body will then stay at this new temperature. 


Option 1 (adjusting) is what we see in living creatures.  Option 2 (accepting the change) is what we see with objects and stuff (that have internal heating).   
