How should I paint around my newly hung doors?
Sounds like someone didn't clean the wall adequately, and/or didn't apply primer, before the previous coat of paint. If it doesn't adhere well, paint can flake or peel off.

However, if there isn't evidence of that the problem is propagating outward, I'd say the answer is just to do the touch-up properly: Clean with a mild degreaser, apply a good quality primer, and paint in matching color. 

(The good news is that you can take one of those torn-off pieces into a paint store and say "match this color"....)