Please use strict and warnings. Please do not use a filehandle called true, then have a while (<true>) that will go false. That hurt my head ;)

It is good practice to test the result of attempts to open a file, and polite to close them when you have finished reading from them.

perltidy is your friend

The three argument form of open is safer

update

Added a check that there is something remaining in the original file while editing

#!/usr/bin/perl
use strict;
use warnings;

# true was a confusing name
open ORIG, "<", "ORIGINAL" or die "probs: ORIGINAL : $!\n";
open EDIT, "<", "cut_out"  or die "probs: cut_out2 : $!\n";
open REST, ">", "REST"     or die "probs: REST     : $!\n";

while (<EDIT>) {
    chomp;
    my $j = $_;
    print "grabbing $j lines to $.out.test\n";
    open OUT, ">", "$.out.test" or die "can not open $.out.test : $!\n
+";
    while ($j--){
        if (my $line = <ORIG>) {
            print OUT $line;
        } else {
            die "Ran out of input file before editing complete !\n";
        }
    }
    close OUT;
}
close EDIT;
while (<ORIG>) {
    print REST;
}
close ORIG;
close REST;

# rename ("REST", "COMPARE");
[download]

Cheers,
R.