It kind of bothers me a lot that there myths about copyright law and fair use that keep getting perpetuated over the years because most people involved in AMV editing don't have a good understanding of copyright law and they've been given incorrect information by others. AMV's do use copyrighted material, and while a lot of people seem to think that they automatically fall under fair use, this is not the case. Fair use is argued on a case-by-case basis. It does not protect you from liability - it is an affirmative defense that you use after you have been brought to court for copyright violation (meaning in order to use it as a defense you have to acknowledge that you used copyrighted material). It is a limitation on the original author's rights, not an exemption from the law.
Fair use is incredibly difficult to argue under most circumstances. While there are guidelines for what qualifies, in the end it will often come down to the judge's opinion on the matter. Let's take a look at some of those guidelines in relation to the anime used in AMVs. The percentage of the original work used is one factor. True, AMVs tend to only use a small percentage of the anime, so you could argue that only a small percentage of the original work was used. Another factor is the benefit AMVs have for the general public. IP (Intellectual Property) used for educational purposes generally falls under fair use, though again, percentages of original work used still are considered in these cases. Parody is another use that actually gets upheld as fair use on a regular basis. However, the majority of AMVs probably don't qualify as parody in the way the courts have defined, and they are definitely not used in schools as an educational tool. Entertainment doesn't really count as public benefit, so the argument for fair use starts to get shaky on this point. That brings me to the issue of profit. I hear the argument a lot that AMVs aren't done for profit, so they are fair use. Surprisingly enough, stuff that makes a profit can be qualified as fair use. Stuff that doesn't make a profit might not be qualified as fair use. A lack of profit off the derivative work might make the judge look on you more favorably, but it is by no means the deciding factor.
Even supposing you do sway the judge on the above arguments with regards to the anime you used, where all of this breaks down completely is the music. You are copying and distributing the artist's work without their permission. Editing the audio in some way does not somehow remove the artist's protection of their IP under the law. Some artists may be ok with it, but what it comes down to is enforcement. Sure, you PROBABLY won't get taken to court, but the RIAA also has a history of making nasty examples of random individuals.
I'm not saying everyone should stop making AMVs, but I think people do need a better idea of the realities surrounding the hobby. The law's not in our favor at this point, and as Bashar pointed out a lot of the behavior on the internet is prompting copyright holders to lobby for stronger laws. In the long run it's not in your best interests to file a counter-claim. Educate yourselves on the law, and if you don't like it, get involved with organizations that are making an effort to change it.
For anyone who wants to know more, here is a decent FAQ that explains copyright law in plain English, so you don't have to struggle through the legalese of the actual statutes. http://www.copyright.gov/help/faq/