Why does the skin on my right index finger peel every 2-3 months?

Contact allergy? This may represent a contact allergy to things which you are touching. Topical cream will suppress reaction. If it is svere patch testing can help identify chemical.
Sympathetic peeling. There is a condition called one hand/two foot disease, which is a fungal infection of the feet (athlete's foot) which left untreated for a long time sets up a reaction in the hand. The fungal infection in the feet may not be bad, or wet and gooey. See your doctor, or a dermatologist for an evaluation.