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

This is a nuanced situation. Becoming a copy of another permanent does not automatically remove counters. According to CR 706.2, the copiable values of a permanent are its characteristics as defined by its text box, power/toughness, and similar printed traits — counters are explicitly not among these copiable values. So the counters physically stay on the permanent even after the copy effect resolves.

However, the creature's base power and toughness change to match the copy's printed values, and then counters (like +1/+1 or -1/-1 counters) are applied on top of that new base. The creature does not 'lose' the counters in the sense that they are removed, but its stats are recalculated from the new base (CR 613.3, CR 706.2).

For example: you have a 2/2 Bear creature token with two +1/+1 counters on it (making it a 4/4). If it becomes a copy of a Grizzly Bears (also a 2/2), it is still a 2/2 Bear token by copiable characteristics, but the two +1/+1 counters remain on it, making it a 4/4 again. The counters survive the copy effect.

The relevant rules are CR 706.2 (what values are copiable) and CR 613.3 (layer system for applying copy effects before counters). A copy effect is applied in Layer 1, while counters affect power/toughness in Layer 7c, meaning counters always apply after the copy effect in the calculation order.