Ok, I studied the code some more and tested and tried and realized the < and > were reversed. This is what I've got now and it seems to work and I think I get it:

while ( my $html_file = $rule->match ) {
     
       rename($html_file, "$html_file.bak")    or die;
       open(my $fh_in,  '<', "$html_file.bak") or die;
       open(my $fh_out, '>', $html_file)       or die;
    
    while (<$fh_in>) {
        my @lines = <$fh_in>;

        for (@lines) {
            #replace <!--#include virtual="[document path]"-->
            #with <?php include($_SERVER['DOCUMENT_ROOT'].'[document p
+ath]'); ?>
                if (s/<!--#include virtual="(.*)"-->/<?php include(\$\
+_SERVER['DOCUMENT_ROOT'].'$1');?>/){
                 my $result = $1;
                 #print the file changed and the document path for the
+ included file
                 print OUTFILE "$html_file: $result\n";    
            }
        print $fh_out $_;
    }

    close($fh_in);
    close($fh_out);
}
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