If a creature becomes a copy of another creature, does it keep its counters?

When a permanent becomes a copy of another permanent or card, the copy effect sets its copiable values — things like name, mana cost, types, abilities, and power/toughness. Counters are explicitly not among the copiable values, so they are unaffected by the copy effect. This is governed by CR 707.2, which lists exactly what is and isn't copied.

CR 707.2 states that the copiable values are the values printed on the card plus any copy effects already applied, and CR 707.2a clarifies that counters, damage marked on a permanent, Auras attached to it, and similar characteristics are not copied. The creature simply retains whatever counters it had before the copy effect was applied.

It's worth noting that the creature's base power and toughness will change to match the copy's printed values, but +1/+1 counters (or any other counters) still sit on top of that, modifying the final result as usual under the layer system (CR 613).

Example: You control a Grizzly Bears (2/2) that has two +1/+1 counters on it, making it a 4/4. You use a spell to turn it into a copy of Runeclaw Bear (also a 2/2). After the copy effect, it is now a Runeclaw Bear with two +1/+1 counters — so it's still a 4/4, just named Runeclaw Bear with Runeclaw Bear's characteristics.

This same principle applies to -1/-1 counters, loyalty counters on a permanent that becomes a planeswalker copy, charge counters, and any other counter type — they all survive the copy effect.