He could be tail biting, but that's if he's agitated. Is there anything you may be doing to make him mad? Mine was tail biting because he didn't like the current from th filter, so I had to make a baffler. If he can see another betta that could be it also.
And here's another thing I think it could be: are there any sharp plants in the tank? Like ones with spines or whatever? Even if they don't seem that sharp, betta fins are a single cell thick, so it doesn't take much to tear them. If so, try just removing it for a while, and see if the tearing stops.