So you made this nice AMV or other video using a nice song, but when you try to upload it to youtube they strip out the music or do whatever youtube does these days. Frustrating, right? There's two ways to prevent this kind of thing: use a different song or use a different website. (Well, I suppose you could stop making AMV's altogether, but where's the fun in that?)
Note that it doesn't say how strict the various sites are on copyright enforcement, but saying “we're gonna be pretty lazy when it comes to that” in such a way that it can be cited on Wikipedia would probably get the site into legal trouble.
Using a different song
There are a number of websites that compile Creative Commons licensed and public domain music. Sadly the one I've been using shut down recently due to lack of funding.
Here's a list of songs I found through the FMA. I will be replacing the links with ones that work soon.
They are a lot less likely to make a fuss. Someone might, if they want to listen to a certain song, watch an AMV of it instead of buying it off iTunes. (As in: I did stuff like that when I was young and stupid.) Nobody's going to watch AMV's instead of buying a box set of the anime. They might, however, watch an AMV of a series they don't know and then decide to check the series out. To the people who own the rights to the anime, AMV's are basically free advertising.