sub nonrepeating {
    my $word = shift || $_;
    my %chars;
    @chars{ split( //, $word)} = ();
    length($word) == keys %chars;
}

my @solutions;
while (<DICT>) {
    chomp;
    push @solutions, $_
        if length() == 15 and nonrepeating();
}
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This uses the short-circuit behavior of and to only do the expensive nonrepeating() test if the word has fifteen characters. As a matter of style, some will object to the $_ default in sub nonrepeating, but I think pronouns fit right in there. This saves some memory over yours at runtime since only actual solutions are pushed.

After Compline,
Zaxo