The following solution only passes once through the dictionary, and doesn't need an additional pass through the hash. It also stores less in the hash. It takes advantage of the fact that the dictionary is sorted, and that a word of the form Xm sorts before Xrn.

#!/usr/bin/perl

use strict;  
use warnings;

my $file = "/usr/share/dict/words";

open my $fh => $file or die;

my %seen;

while (<$fh>) {
    chomp;
    if (/m$/) {
        $seen {+lc} ++;
        next;
    }
    elsif (/rn$/) {
        if ($seen {lc "$`m"}) {
            print "$`m/$_\n";
        }
        next;
    }
}
__END__
bum/burn
chum/churn
dam/darn
hom/horn
modem/modern
stem/stern
tom/torn
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