Your file isn't changing for each pass of the while loop, so you can open it prior, and close it after.
# note the three-arg use of open
open my $fh, '+>', 'results.txt' or die "Can't open results.txt: $!";
while ( $html =~ /"resumeDetailLink" href="([^"]+)"/sg ){
my $link_url = 'http://www.beyond.com' . $1;
print $fh $link_url . "\n";
}
close $fh;
UPDATE: I forgot to point out that the reason it didn't work originally was because you forgot to pass the file handle as a parameter to print(). Had you of done that, it would have worked. However, because you would have been opening and closing the file upon each iteration of the while loop, it would have been very inefficient.
