I never get burned out on the anime I use. It's worth mentioning though that since I started watching anime, I've done very little re-watching of any of it. That's partly because the amount of anime I've watched is so pitifully small, and until recently it took a lot for me to sit down and watch through anything, anime or otherwise. But I can say that editing hasn't been the reason for my hesitance in rewatching any anime.
Music, though. That's a whole other can of worms. Generally, by the time I get through editing a video, no matter how much I enjoyed a song at the beginning, I never want to hear it again at the end. I have an extremely low tolerance for listening to songs over and over again. I just can't do it. I know people who can listen to the same song on repeat and enjoy it just as much each time they hear it. I find it difficult to listen to the same song more than once in a day.
For me this is one of the single most frustrating things about the entire hobby. When I listen to music, especially with songs I like, because I'm an editor I naturally try to pair the music up with visuals in my head and I start forming video ideas automatically. So many ideas I've had to just let go unrealized, though, because my love for the song is too great. The cost of making a great video is ruining a song I love -- nope, not worth it to me. For this reason, several bands I love are completely off-limits to me. I will never make a video to a song by Brand New, Belle and Sebastian, LCD Soundsystem, M83, or several other artists even though they're some of my favorites.