if you think it's infected or it reddens, get it checked at pharmacy or doctor
Mayo Clinic:
Minor hand and finger burns
Minor burns:
are smaller than 3 inches
cause superficial redness
make blisters form
cause pain
don’t break the skin
Minor burns require immediate action but often don’t require a trip to the emergency room. You should:
Run cool water over your finger or hand for 10 to 15 minutes.
After flushing the burn, cover it with a dry, sterile bandage.
If necessary, take over-the-counter (OTC) pain medication such as ibuprofen (Advil, Motrin), naproxen (Aleve), or acetaminophen (Tylenol).
Once it’s cooled, put on a thin layer of a moisturizing lotion or gel such as aloe vera.
Minor burns will usually heal without additional treatment, but if your pain level doesn’t change after 48 hours or if red streaks start spreading from your burn, call your doctor.
Things not to do for finger burns
When performing first aid on a burned finger:
Don’t apply ice, medicine, ointment, or any household remedy — such as butter or oil spray — to a severe burn.
Don’t blow on a burn.
Don’t rub, pick, or otherwise disturb blistered or dead skin.