local $/;

You have this at the top level. I think it's more clear to say undef $/;. The form using local is good if you're doing this inside a block and don't want to meddle with the value outside your block.

open(FH, $file) or die "File couldn't be opened";

I'd say:

open my $fh, '<', $file or die "Can't read $file: $!";
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The three argument open saves you from some odd cases, and the error message including the filename and $! tells you a lot more about what failed and why.

        my $start = $1; $start--; #increment down for substr
        my $end = $2; $end--; #increment down for substr
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I'd move those lines up to immediately after the pattern match and keep them there. If you add some intervening code later that uses a pattern match, these won't work. Also, these expressions might be clearer as, for example, my $start = $1 - 1 instead of the separate decrement.

Your html_out would be better off accepting a hash reference instead of trying to stuff the whole hash in as arguments. You'd call it as html_out($total_mutations, \%mutant_genes) (note the backslash), and it would take its arguments like so:

sub html_out {
    my ( $total_muts, $mutant_genes_ref ) = @_;

    my %mutant_genes = %{$mutant_genes_ref};
    # ...etc...
}
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The rest of the code wouldn't change. I don't think it's a big deal in this case since %mutant_genes is global anyway (by_descending_probability is already using it). Have a look at reftut for more about references.

On the plus side, I found it pretty readable (though I ignored the big hunks of HTML), and the comments are good.