If you're using Perl 5.005_03 or higher, this is how I'd write your code:

open my $results, '<', $outputfile
    or die "Cannot open '$outputfile' for reading: $!\n';

open my $cleaned, '>', $new_outputfile
    or die "Cannot open '$new_outputfile' for writing: $!\n";

{
    my %seen;
    local $_;
    while ( <$results> ) {
        chomp;

        next if $seen{ $_ }++;
        print $cleaned $_, "\n";
    }
}

# Always close in the reverse order of opening.
close $cleaned;
close $results;

unlink $results
    or die "Cannot unlink '$outputfile': $!\n";
File::Copy::move( $new_outputfile, $outputfile )
    or die "Cannot rename '$new_outputfile' to '$outputfile': $!\n";
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The changes:

• Use 3-arg open for safety.
• Use lexicals for filehandles instead of globals. (That's the difference between 'my $fh' and 'FH'.)
• Always check the return values of system commands.
• If you fail a system command, then stop processing immediately. It's dangerous to continue unless you have a way of cleaning up what went wrong.
• Don't read the file into memory. It might be small now, but it will probably grow.
• If you intend on using $_, localize it so that you don't trample on anyone.
• You're not paying for your whitespace, so use whitespace liberally. It will increase readability in the long run.
• If you open it, then close it as soon as you're done with it.
• If you open two things, close them in reverse order.