use strict;
use warnings;

while (my $line = <DATA>) {
    if ($line =~ /#include [<"](\S+\.h)[>"]/) {
        print "$1\n";    #not printing the header third case
    }
}

__DATA__
#include "string.h"
#include <stdlib.h>
#include <sys/dispatch.h>
[download]

Prints:

string.h
stdlib.h
sys/dispatch.h
[download]

which is what I understand you to want, so where's your problem?