Anything flammable will burn if it gets hot enough. How hot depends on the particular substance and other factors such as the air pressure and how much oxygen is around. 100% cotton can auto-ignite if it gets hot enough.
There's too many factors (density, how it was woven, if there's any oil/grease on it) to say definitively what that temperature is, but in general I wouldn't put cotton in the microwave or in the oven.
Similarly, if you get any kind of oily stain on clothing, you shouldn't put it in the dryer! That's one of the ways you can get a dryer fire. The auto-ignition temperature of the cloth is lowered by the oil (yes, even vegetable oil).