Not only can this be made much shorter and more efficient, your second example does not actually find all valid matches. Rewritten, with only one pass through the dictioanry:

#!/usr/bin/perl -w
use strict;

open my $dict, '<', '/usr/share/dict/words'
  or die "open failed: $!";

my %words;

while (<$dict>) {
  chomp;
  next unless s/(?:m|rn)\z//;
  ++$words{$_};
}

while (my ($word, $count) = each %words) {
  print "${word}rn / ${word}m\n" if $count > 1;
}

__END__

darn / dam
morn / mom
yarn / yam
churn / chum
modern / modem
torn / tom
burn / bum
stern / stem
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