my @rules; # not %rules.
...
# Process input file of processing rules
while(<INFILE>) {
   ...
   push @rules, {
      regex       => qr/$string/,
      file_handle => $fh
   };
}

...

# Read Apache log file and print to various other files
while (my $line = <STDIN>) {
   for my $rule_ref (@rules){
      my $regex = $rule_ref->{regex};
      my $fh    = $rule_ref->{file_handle};

      if ($line =~ $regex) {
         print $fh $line;
      }
   }
}