If I sat in a perfectly insulated box, what would happen? We have a perfectly insulated box.
What if I sat inside the box. My body acts as a heat source, right? Would the temperature inside the box increase all the way up until I die? Or, will an equilibrium eventually be reached (maybe body heat production varies based on surroundings?)
Let's say I bring an unlimited amount of food and water. Would I eventually burn to death?
 A: Yes, you would die of overheating, but you would not burn, not even in the slighest manner, in the sense that paper burns. If the box was perfectly insulated, once your core temperature rises above the levels below, you would not be able to maintain Homeostasis.
There is some interesting physics in these related articles, one regarding the production of excess body heat by bees as a method of protection and the other short article about surviving in an igloo in polar regions.
How honey bees kill hornets that invade their nests
Body heat in an Igloo

Inuit people tended to use snow to insulate their houses, which were constructed from whalebone and hides. Snow is used because the air pockets trapped in it make it an insulator. On the outside, temperatures may be as low as −45 °C (−49 °F), but on the inside the temperature may range from −7 °C (19 °F) to 16 °C (61 °F) when warmed by body heat alone.

From How does heat kill you

In general, your nervous system starts working to cool you down every time your central temperature rises a couple of degrees above normal. To do that, it tries to divert blood away from your internal organs and toward your skin. Since blood carries a lot of heat, your best chance of cooling off is to get that hot stuff pumping to the surface. Meanwhile, your sweat glands start to release water, which has a cooling effect as it evaporates from the surface of the skin.
You may not be able to radiate any heat from the surface of your skin, and your sweat won't evaporate fast enough to keep you cool.  The heart responds by pumping more blood away from the internal organs. This deprives the intestines of oxygen, which damages their linings and makes them more permeable to endotoxins. At the same time, an overheated core causes an inflammatory response throughout the body. The combination of the inflammatory response and the endotoxins in the bloodstream can suppress the body's natural mechanism for cooling down.
Once your core gets above about 104 F degrees, you're in serious danger. High internal temperatures lead to increased pressure in your skull and decreased blood flow to your brain.Damaged tissue may also enter your bloodstream and lead to kidney failure. Very high internal temperatures—like 120 F degrees—can destroy the cells in your body through direct heat damage.

A: We actually have a one cubic meter styrofoam box in the course lab. Students can volunteer, their heat output is around 80 watts (we only leave them in there for five minutes). When CO$_2$ and O$_2$ is no problem, I believe overheating would become dangerous in about one hour. Compare with the rate at which bodies cool down after metabolism ceases. You won't need food in there.
