What you can do to prevent anything from beeing to STDERR is close the file descriptor :

print STDERR "prints...\n";

close(STDERR);

print STDERR "doesn't print...\n" or print "this was bound to fail\n";
[download]

But any entity that tries to write to STDERR will fail doing that, so you should also reopen STDERR to /dev/null to avoid that.

open(STDERR,'>/dev/null');

print STDERR "doesn't fail so this..." or print "...won't print";
[download]