Your biggest problem is in the last while(<FILE>) loop: you're trying to read from a file that you just opened for writing.

The other problem is with the grep: you're only collecting lines that start with a #, although you say you want to do the opposite.

You seem to be doing a lot of unnecessary looping too. I suggest the following as a more streamlined alternative:

use strict;
use warnings;

print "Input file name? ";
my $name_in = <>;
chomp $name_in;

print "output file name? ";
my $name_out = <>;
chomp $name_out;

open my $in, '<', $name_in or die "error opening $name_in: $!";
open my $out , '>', $name_out or die "error opening $name_out: $!";

print $out grep { !/^#/ } <$in>;
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