I know each $id in @ids does occur in one of the files but in my code the result of print is always empty!

The quickest thing to try is to chomp your @ids:

my @ids =<IDLIST>;
chomp @ids;
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You should not assume that one approach is faster than another; Benchmark it instead. Avoiding unix grep should make your code more portable, but you would have to measure whether it is faster with or without unix grep.

Also, I think you are unnecessarily looping through all your files multiple times. It seems there is no need for your foreach my $fasta (@files) { loop.

Here is a slightly more conventional approach, in my opinion, which does not use Perl's grep:

foreach my $id (@ids) {
    foreach my $fasta (@files) {
        open my $fh, '<', $fasta or die "can not open file $fasta: $!"
+;
        while (<$fh>) {
            print if /$id/;
        }
        close $fh;
    }
}
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