Can an affected tooth nerve heal by itself?

Sometimes: a tooth nerve that is sensitive to cold can heal itself. This is called reversible pulpitis. If the nerve is sensitive to,not or decay has progressed into the nerve then it cannot heal itself and needs to be treated by a dentist.

Up to a point...: For example, if you had a moderately deep restoration placed and the nerve was affected by the procedure and became inflamed, then yes, it can "repair" itself and the sensitivity should subside. If the nerve is infected by bacteria or irreversibly inflamed, then no, the tooth would need root canal therapy.

Sometimes: There are reversible and irreversible inflammations of the nerve. Your symtoms and x-ray will tell your doctor which direction the nerve is going.

Reversible pulpitis: If the nerve is exposed to decay it becomes inflamed and then infected. Teeth should not hurt to bite on and should not hurt by themselves spontaneously. Reversible pulpitis may Initiated by a stimulus i.e. cold or hot and short duration i.e. 30 seconds or less

Sometimes: Often, the tissue inside the tooth (the pulp) can calcify over time, and the nerve canals actually end up having a natural root canal. The tooth will not be vital, but is normally OK as far as other functions. I would have dentist keep m monitoring the tooth.