Does a creature that becomes a copy of another creature retain its counters?

When a creature becomes a copy of another creature, the copy effect only changes the permanent's copiable values — things like name, mana cost, color, types, abilities, and power/toughness. Counters are not copiable values and are therefore unaffected by the copy effect. This is established in CR 706.2, which lists exactly what is and is not copied.

CR 706.2 explicitly states that counters on a permanent are not copied. The permanent keeps whatever counters were already on it before the copy effect was applied. This is true regardless of what kind of counters they are — +1/+1 counters, -1/-1 counters, loyalty counters, or any other type.

It's worth noting that the creature's base power and toughness will change to match the copied creature, but any +1/+1 or -1/-1 counters continue to modify that new base stat. So a creature with three +1/+1 counters that copies a 2/2 becomes a 5/5, not a 2/2.

Example: You control a Grizzly Bears (2/2) with two +1/+1 counters on it, making it a 4/4. You use a Clone-type effect to make your Grizzly Bears become a copy of a Runeclaw Bear (also 2/2). Your permanent is now a Runeclaw Bear with two +1/+1 counters — it is a 4/4 named Runeclaw Bear, not a 2/2.