The problem is that each time you append the new line to $outdata, and then run your replacements against the entire text (not just the newly added line). To fix it, either run your replacements once outside of the loop or run them only on the new line. Actually, I would suggest another approach without the loop.

  sub find_replace {
    my $filename = shift;
    open (my $infile, "$dir/$filename") or die "Can't open file: $!";
    local $/ = undef;
    my $outdata = <$infile>;
    close $infile;
    $outdata =~ s{../CLRIS/}{}g;
    $outdata =~ s{../../menu/}{}g;
    $outdata =~ s{../../Images}{Images}g;
    $outdata =~ s{<cfmail}{<!--- <cfmail}g;
    $outdata =~ s{</cfmail>}{</cfmail> --->}g;
    open (my $outfile, "+>$dir/$filename") or die "Can't open file: $!
+";
    print $outfile "$outdata";
    close $outfile;
}
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Oh, while I'm looking at it, I suspect you don't actually mean s{../CLRIS/}{}g;. It seems much more likely that you mean s{\.\./CLRIS/}{}g;