Do I have to pay duty when carrying 15 iPhones domestically in India? (From Chennai to Delhi.)

No, you don't have to pay duty fees when carrying cell phones from Chennai to Delhi. Both cities are in India, India is a unified federal republic, and import taxes are extremely rare between states. There are apparently some import procedures related to vehicles and some special goods into a small minority of states, but I don't believe this applies for bringing such a small number of cell phones to Delhi. Such laws are likely made for major retailers and big businesses. 15 is a relatively small number, bordering on personal use. They could all be gifts to friends and family.

I have crossed state borders in India many times, both through bus and through plane. I don't recall ever having seen any domestic customs checks. I don't even know how it would be possible to find facilities to try and declare something after a domestic flight or while crossing state borders by road.