Minor stylistic nitpicks before anything substantive: lexical filehandles are better than bareword globals like you've used, and you want to include $! in the error message so you know why the open failed.

open( my $ips, '<', 'TEST_IPS.txt' ) or die "Can't open IPS: $!\n";
[download]

That out of the way you're close but you don't consult %interesting_ips in any way to tell if you have a match. You also want to pull the list of matching IPs out differently.

#!/usr/bin/env perl
use strict;
use warnings;
use Regexp::Common qw( net );

open ( my $ips_fh, '<', "TEST_IPS.txt" ) or die "can't open IPS: $!\n"
+;
open ( my $rules_fh, '<', "test_rules.txt" ) or die "can't open rules:
+ $!\n";

my %interesting_ips;

while( <$ips_fh> ) {
  chomp;
  $interesting_ips{ $_ } = 1;
}

close( $ips_fh );

while( my $fw_line = <$rules_fh> ) {
  chomp( $fw_line );

  my( @addresses ) = $fw_line =~ m{ ($RE{net}{IPv4}) }gx;

  for my $addr ( @addresses ) {
    next unless exists $interesting_ips{ $addr };

    print qq{$.:interesting '$addr': $fw_line\n};
  }
}

close( $rules_fh );

exit 0;

__END__
$ perl pm_foo.plx
1:interesting '10.198.0.0': 133 bba33132-6192-51e8-4d78-c1b7bfd47251 a
+ny V072-AklC-DB MOSSACSQLAdminGroup 10.198.0.0/16 10.210.0.0/16 MOSS_
+SQLAD_10-208-22-1/28 accept  always RDP TCP17338 TCP18230 PING all   
+   change 125213      157309
2:interesting '10.198.1.0': 136 a5ea4ee8-6192-51e8-0252-2017208af83d a
+ny V071-AklC-Web ACHendersonRDPUsers 10.198.1.0/16 10.210.0.0/16 MOSS
+_InternalWeb_10-208-22-16/28 accept  always FTP all
[download]