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

When a permanent becomes a copy of another permanent or card, the copy effect only changes the permanent's copiable values — things like name, mana cost, color, type, subtypes, abilities, and power/toughness. Counters are not copiable values and are therefore unaffected by the copy effect.

This is established by CR 706.2, which lists exactly what is included in copiable values, and by CR 706.3, which clarifies that copy effects do not affect counters, which are considered part of a permanent's physical state rather than its characteristics.

It is important to note that the creature's base power and toughness will change to match the copy's printed values, but any +1/+1 or -1/-1 counters (or other counters) that were already on the creature will still be there, modifying its stats on top of the new base values.

Example: You control a 2/2 Grizzly Bears with two +1/+1 counters on it (making it a 4/4). It then becomes a copy of a Runeclaw Bear (also a 2/2). After the copy effect, it is a 2/2 Runeclaw Bear with two +1/+1 counters still on it, so it remains a 4/4 in terms of actual power and toughness.

This same principle applies to all counter types — loyalty counters, charge counters, -1/-1 counters, etc. None of them are removed when a permanent becomes a copy of something else.