What could cause burning under the skin in the armpit?

Underarm Dermatitis. Burning in an armpit can have several causes. You may be irritating it because of your deodorant. It's wise to see a dermatologist and see if you have a contact dermatitis or some more esoteric infection such as Candida Albicans or a fungus on the hairs.
Irritation. It could be an infection or excessive sweating that's causing irritation. It could also be a reaction to chemicals in clothing. It it is occurring on only one side it may a symptombof something thatvis unrelated to skin from deep inside. If it persists it is best to be evaluated.