What could cause this burning sensation under my index fingernail?

Finger tip pain: Pain in your finger tips can come from many sources. If your nail looks normal and is not tender, there may be pain coming from either a nerve problem (neuropathy) or a vascular problem (such as raynaud's syndrome). Repeated minor trauma (from tapping your finger) can sometimes cause this too.

Glomus? : There is a tumor called a glomus tumor. This is not cancer it is benign. It causes pain and cold insensitivity of the finger tip. Treatment involves removing the tumor.