There are a few more with (missing) "g" modifier; and s/5/6/; anyway it gets above 1 s for me. OTOH (definitely not a one-liner though):

#          Rate am_new
# am_new 40.9/s     --

    am_new => sub {
        my ( @a, %h );
        open my $fh, '<:raw', 'unixdict.txt' or die;
        while ( <$fh> ) {
            next unless length > 6;
            chomp;
            push @a, $_ if -1 != index $_, 'e';
            next unless -1 != index $_, 'i';
            my $key = tr/e/i/r;
            $h{ $key } = $_
                unless exists $h{ $key }
                    and $_ lt $h{ $key }
        }
        close $fh;
        my ( @ret, $key );
        exists $h{ $key = tr/e/i/r } 
            and $h{ $key } gt $_
                and push @ret, sprintf "%30s  %s\n", $_, $h{ $key } 
                    for @a;
        @ret
    },
[download]

And, e.g. for (some or all of) "e => a" (not "i") then "gt", "lt" should be swapped.

Oops, yes, missing g. But did they change the problem? I see "The length of any word shown should have a length > 5."

$h{ $key = tr/e/i/r } is wrong since $_ not set there. Perhaps the = should be a =~?

(Nevermind)