I didn't wear gloves yesterday and i got frostbite. there was this burning sensation when i got warm. my fingertips are still tender is this normal?

Normal for injury: You should take care to avoid additional cold injury as your hands heal. Unless you have blisters or blackened skin the fingers should heal.

Yes: It take few days to get over it unless you have permanent damage to the tissue or the nerve of the skin.

Yes: Hopefully the sensation will recover and the burning stop. Reheating after frostbite induces a reaction hyperemia (increased blood flow). Ghte neves are the most metabolically active tisue in your fingers so they "burn" as they -hopefully recover.