Tapping and dragging seems intuitive enough, but it would be nice to see some UI change when you've tapped (maybe have the disk light up,) and maybe an option for tapping twice instead of dragging as well.
Overall presentation could use some work. You have a nice 3D model of the towers in the menu, why not do something similar for the game itself?
As for the game itself, there's not much to say. The game gets pretty long later on and I never seem to get the parity right (totally my fault, I know) but similar to a Rubiks cube, it's less of a puzzle game and more of a procedure once you know the deal. Then again, Tower of Hanoi has been around forever and it's not like you're going to fuss with an old formula and still call it the same thing.
Decent attempt but ultimately kind of boring. I can't imagine anyone would bother solving a 15 disk tower. (Seriously??)