This solution also depends on a dictionary, but it doesn't read the whole thing into memory. It tests each line of it as it reads through.

my $dictionary = '/usr/share/dict/words';

# input letters, changed to lowercase
my @letters = split //, lc shift @ARGV;

open my $dict_fh, '<', $dictionary
    or die "Can't read dictionary '$dictionary': $!";
WORD:
while ( my $word = lc <$dict_fh> ) {
    chomp $word;

    # $word is lowercase with no newline
    my $orig_word = $word;

    # for each letter in the search set,
    for my $letter ( @letters ) {

        # remove that letter from the dictionary word
        # and skip to the next word if it's not found
        next WORD if ! ($word =~ s/$letter//);
    }

    # if all the dictionary word's letters were used,
    # print out the original dictionary word.
    print "$orig_word\n" if $word eq '';
}
close $dict_fh or die "Can't close: $!";
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Update: Added some comments to describe the algorithm (thanks to Limbic~Region for the suggestion). Also, this code is tested (with the case the OP specified).