The other solutions listed here are better, but if you really want to do it in your perl script:

First way: ./script | ./monitor.pl

# monitor.pl
while(<>) {
   if(/wantthis/) { dosomething($_); }
   if(/warning/) {dosomethingelse($_); }
   print; ## pass on the output
}
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Option 2 - run the program from perl

# monitor.pl
open(INPUT, "./script |") or die $!;
while(<INPUT>) {
  # same as above
  if(/regex/) { dosomething($_); }
  print;
}
close INPUT;
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Yet another way: If the script itself is perl, have it monitor it's own output.

# adapted from Perl Cookbook
sub filteroutput {
   return if my $pid = open(STDOUT, "|-");
   die "cannot fork: $!" unless defined $pid;
   while(<STDIN>) {
     if(/regex/) { dosomething($_); }
     print;
   }
   exit; #exit forked monitoring process
}
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Now add filteroutput(); above the rest of the script's normal code. It's self-monitoring.