I guess that burnig sensation is supposed to feel "cooling" and "refreshing", must be the menthol or some other stuff.
However if you have dry skin, it can be too irritating, and in winter, especially if it is -20 degrees celcius and the air is very dry and makes your skin dry too, you should avoid irritating your skin more than necessary.
I mostly shave without any foams, in the shover, or use just some plain moisturizing cream instead of shaving foam. Shaving in sauna is also very comfortable as the heat softens your hair, that's where I shave my balls.