One synonym for "keep one's hands off" is "refrain." When someone refrains from touching or interfering with something, they are keeping their hands off it. Another synonym is "abstain," which means to voluntarily avoid doing something. "Avoid" is also a synonym, because it means to stay away from a particular action or thing. "Resist" is another word that can be used, as it implies exerting self-control to not touch or manipulate something. "Refrain from meddling" is also a similar phrase. All of these synonyms convey the idea of intentionally not touching or interfering with something.
