I've experienced both burnout and no burnout. It really sort of depends on the situation. I also don't rewatch stuff too often (nearly at all; it took over 10 years before I actually rewatched the Ah! My Goddess OVA), but that's because I have a really uncomfortable setup and so my anime watching energy generally is always focused on watching a series for the first time.
With music I may listen to something over and over again right after first hearing it, but then it tapers off. If I make a video to a song, I may get burned out on it during the process of editing, but once it's finished, that fatigue dissipates until I tire of watching the video over and over again, and then it may take a few months before I can stand to listen to the song again. But the longer time passes, the less burned out on something I get.
Other times, the editing process was so short that I didn't have a chance to get burned out on the music. This generally happened more during the early part of my editing hobby, with the videos I'm generally embarrassed by and edited in under a week (often, less than a day or two). There's only been a couple instances where making a video made me never want to hear the song again (I cringe when I hear Powerman 5000's "When Worlds Collide", although I'm fine with the rest of their stuff, more or less).
More recently, the editing process (and watching the finished result) can make me more aware of the nuances in a piece of music, and that can help greatly in shortening the burnout period.