My experiences with burning hot foot were due to cycling shoes that were just a smidge too narrow. It would seem ok for a while, but eventually lead to excruciating fire. And yet, as many other posters here have noted, too much foot movement in your shoes can also cause it.

Basically, a proper fitting shoe should be:
1. Wide enough in the toebox to not pinch nerves (Hello, Morton's!) but not so wide that you can shift your feet from side to side.
2. Narrow enough in the heel to grip it firmly.
3. Very stiff sole, as a flexible sole will cause your foot to flex around a lot triggering irritation and inflammation (see next point).
4. Some sort of footbed that properly cradles your foot. Arch support that is stiffer - more stiff than you'd ever want to walk in- is important for many people because it holds the foot rigidly and does not allow it to flex excessively. Without arch support, your metatarsal area splays out wide as your press your foot down and then shrinks back as you pick your foot up. This constant motion within your foot causes a lot of friction on the nerves and muscles between the bones. If the situation continues long enough, you can almost form something like an internal blister.

Aside from shoes, pay attention to your physiology. Some amount of swelling during exercise is common, but edema is something you need to watch out for. If you notice an indentation from your sock elastic after a ride, or puffy ankles that you can make a thumbprint mark on, then you've got edema. Make sure you're properly hydrated and getting enough replacement electrolytes in your diet. If it persists, see a doctor.