As unpopular as this is likely to be... AMVs at US-based anime conventions are fairly flagrant copyright violations. They are clearly derivative works, using the artwork of an anime to either summarize the story or create a different story from the original work.
The American copyright law summarizes this fairly clearly:
A “derivative work” is a work based upon one or more preexisting
works, such as a translation, musical arrangement, dramatization,
fictionalization, motion picture version, sound recording, art
reproduction, abridgment, condensation, or any other form in which a
work may be recast, transformed, or adapted. A work consisting of
editorial revisions, annotations, elaborations, or other modifications
which, as a whole, represent an original work of authorship, is a
“derivative work”.
(On a side note... yes... this is the reason fansubs are violations as well)
Unlike what several people here have said, copyright is NOT limited to civil law (the type of law where the copyright holder is the one who has to bring a case against the offender). Both the Copyright Act and the DMCA have established criminal penalties for cases of 'willful copyright infringement'. This does mean that it is possible for law enforcement to investigate, arrest and prosecute copyright violators without the involvement of the copyright owners. From a practical perspective, this would be very difficult, since law enforcement needs to know if some type of permission has been given. (and this type of investigation is usually limited to mass importers of bootleg DVDs/CDs)
Realistically speaking, however, anime music videos are fantastic advertising for the copyright owners property. We are unlikely to see any type of hostile action against them at any time.
A more realistic problem is the other media being flagrantly infringed upon in an AMV. Unlike the anime, which is typically sliced into small "beat-sized" fragments and rearranged at the whims of the creator, the MUSIC part of the AMV is typically a straight copy of a recording. This violates the musicians song copyright, the recording company's media copyright, AND the public production copyright (owned by who-knows).
Once again, if the song is from the anime, any prosecution is unlikely. For popular American pop songs, however, anime conventions would be well advised to look into obtaining blanket licenses from the major clearing houses. (the same operations that sell permission to bars/nightclubs for 'public performance')