How do other primates treat their fingernails?
Both. Many primates bite their nails. Some do not. It's not species specific.

Chimpanzees, for example, usually bite their nails; some prefer to leave them alone. They do grow continuously, and nails not kept trimmed probably break off (looking at the thickness of those nails, I get the impression that this might be rather painful if they break too short.)

The caption for the picture below (taken from a chimpanzee sanctuary) states


  Annie likes to pick Missy’s nose. Missy tolerates it but doesn’t seem to enjoy it. I don’t blame her – Annie doesn’t bite her nails down like the other chimps do.




They bite their toenails as well.