Do all primates have to trim their fingernails in some way, or do some primates' fingernails wear off through natural use?
Also, is constant nail growth common to all primates?
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.
Why some human beings bite their nails? Because, they are primates - it is inherited way to treat our nails, called onychophagia. While normal in some primates (and this is the way they treat their nails), it is considered abnormal in human.