Word for listening to a single song over and over
You are wearing it out; also the song itself could be wearing thin.
If you loop a song, you play it on repeat. (You can also put it on repeat.)
This doesn't necessarily have the connotation that the original phrase seems to have of wearing out the song's welcome. To clarify, you can use an adverb like endlessly.
There are other phrases, not specific to music, about repeating an action until it becomes boring/obnoxious.
In case somebody likes it, another one I found on the Internets yesterday was "to burn the song into your brain"...