Do -1/-1 counters placed by persist cancel out +1/+1 counters already on the creature?

When a permanent has both +1/+1 counters and -1/-1 counters on it at the same time, the game immediately removes pairs of each until only one type (or neither) remains. This is a state-based action defined in CR 704.5q, which runs continuously whenever a player would receive priority.

Persist (CR 702.79) returns a creature from the graveyard to the battlefield with a -1/-1 counter on it. If that creature already had +1/+1 counters on it before it died, those counters are gone — counters don't survive the zone change from battlefield to graveyard (CR 400.7). So the -1/-1 counter from persist won't cancel counters the creature had in a previous life.

However, if something adds +1/+1 counters to the creature after it returns via persist (while the -1/-1 counter is already on it), the cancellation rule kicks in immediately. One -1/-1 counter and one +1/+1 counter annihilate each other, leaving the creature as if neither existed.

Concrete example: Your Murderous Redcap dies and returns via persist with a -1/-1 counter. You then cast Dromoka's Command to put a +1/+1 counter on it. State-based actions fire: the one +1/+1 counter and the one -1/-1 counter are both removed. Murderous Redcap is now back to its printed stats with no counters, and — crucially — it has no -1/-1 counter, so persist is ready to trigger again if it dies.

This interaction (often called the 'persist combo') is a well-known and intentional rules interaction used in tournament play.