It is a metaphor in this example. Chafing is rubbing which causes heat or irritation and in this case the unstated comparison is with a dog or similar creature struggling against (for example) leather restraints in a desire for freedom. Sonny is looking for some way to escape Don's control.
Edit: in fact, it's really a mixed metaphor as Sonny is said to be under Don's thumb. However, both phrases are really clichés: metaphors that have lost their power by constant repetition.