I think any song can be overplayed, but the threshold for that overplay depends on the music.
There are a few symphonies that I could listen to thousands of times and still find something new in them.
Example: I played Dvorak's "New World Symphony" in school, had a couple of LPs and CDs of the symphony (I had to find interpretations that I like in both formats). I've heard the symphony hundreds of times.
One day I'm listening to the 4th movement and I'm hearing a part where he took parts of two main themes, spliced them together to make the melody, another main theme for the bass line, and a fragment of another for the counter melody. So he has the main themes from all 4 movements stitched together for a short time, and it worked. It was done so cleverly that it took hundreds of joyful listenings to discover that.
On the other hand, a one-chord chant type song gets old pretty quickly.
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And regarding wrong notes. Yes, by all means play it again next time around. Any listener will think you did it on purpose. Some will think it's a poor choice, and other will think you are bringing out your inner Thelonius Monk.

And you are always a half step from a note that works, so if you are quick enough, resolve it a half-step higher and it was just a resolved tension.
That works for jazz, rock, pop, country and other improv art forms.
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One more thing on over exposure (to get back on thread). There are good songs that were overplayed when they were new, and then we all seemed to tire of them. After some time of not hearing the song, if it's played again, I enjoy it again. It's like visiting an old friend.
Insights and incites by Notes