Antonyms for the phrase "keeping an eye on" could include neglecting, disregarding, ignoring, overlooking, and abandoning. These words indicate a lack of attention or concern towards something or someone, which is the opposite of actively monitoring or observing them. A person who neglects or disregards something may fail to notice changes or issues that could have otherwise been prevented by paying attention. Similarly, ignoring or overlooking something could lead to problems going unnoticed until they become significant. Abandoning something or someone implies a complete lack of interest or involvement, which also contrasts with the idea of keeping an eye on something.