mv complains about standard input in a pipeline that filters file name with grep

Your problem is the -L switch to grep. From the man page:

L, --files-without-match
Suppress normal output; instead print the name of each input file from which no output would normally have been printed. The scanning will stop on the first match.

So instead of listing the file names that do not match avi it only lists the place where grep didn't find avi. Since grep usually looks in a file, such a switch would list file names not containing the pattern avi within them. In your case, however, since you're piping to grep from find, the file that grep is reading is (standard input). To see what I mean, try:

find /media/mathscd | grep -L "*.avi"

result:

(standard input)

In other words, grep found lines not matching "*.avi" on standard input, which is exactly what happened.

Since find lists file names directly, you only need to invert the sense of the match:

find /media/mathscd | grep -v ".*\.avi"

(note that grep patterns are different from shell wildcards; you also don't need the -r switch)

Also, using find alone:

find /media/mathscd ! -name "*.avi"