You are almost there. Just "print". Then output to whatever file you want using  yourscript.pl > outputfile.txt

#!/usr/bin/perl -w
print "File name to renumber:";
my $file = <STDIN>;
open (FILE, "$file") || die $!;
$|++;
my $line = 0;
while (<FILE>) {
    next if /^%/ ;
    s/^(?:N\d+\s?)?(?{$line++})/N$line / ;
    print;
}
close (FILE);
[download]

You might also want to use these switches in your command line:

 -i[extension]   edit <> files in place (makes backup
                 if extension supplied)
 -n              assume 'while (<>) { ... }' loop around program
[download]

If you call your script with  perl -i -n all you need is:

s/^(?:N\d+\s?)?(?{$line++})/N$line / unless /^%/;

update: thanks Arien