Does damage dealt to a creature with persist remove the -1/-1 counter placed when it returned?

Damage and counters are completely separate game concepts in Magic. When a creature with persist dies and returns to the battlefield, it enters with a -1/-1 counter on it (CR 702.79a). That counter permanently modifies the creature's power and toughness until something specifically removes it — damage does not do that.

Damage dealt to a creature is tracked as a separate value and only matters for determining if a creature has lethal damage (CR 120.3). It is removed at end of turn during the cleanup step (CR 514.2). Counters, by contrast, remain on the creature indefinitely until an effect explicitly removes them (CR 122.1).

The only ways to remove a -1/-1 counter are through specific card effects — for example, Fate Transfer, Hapatra, Vizier of Poisons-style interactions, or proliferate working in reverse via effects that remove counters. Simply dealing damage accomplishes nothing toward removing counters.

Concrete example: Your Woodfall Primus (5/5 with persist) dies, returns with a -1/-1 counter, becoming a 4/4. Your opponent deals 3 damage to it. The creature is now a 4/4 with 3 damage marked on it — the -1/-1 counter is still there. At end of turn, the 3 damage is removed, but the counter remains, and Woodfall Primus stays a 4/4.