This is a sentence from Duolingo. The app translates it as "don't stay with him". Does this mean don't stay at his place or break up with him?
It can mean both depending on the context. But it more often means break up with him. It would mean don't stay at his place if someone had explicitly asked about staying with him in a specific situation e.g. staying in his flat for this night. The (warning) answer could be Nein, bleib nicht bei ihm. But this is not as often.
To express the request to leave the place next to him it's common and natural to add a verb, which describes the action the addressee does in order to keep the place, e.g. Bleib nicht bei ihm stehen if the addressee stands next to him or Bleib nicht bei ihm sitzen if the addressee sits next to him.