...What I want is just to get these results to be saved in a new text file. ..
That can easily be done by printing your desired output to a file filehandle, you have opened like so:

use strict;
use warnings;

# please note the 'sign' ">"
open( my $fh, ">", "output_file.txt" ) or die "cannot open output_file
+.
+txt$!";
open( my $file, "<", "/file.txt" ) or die "cannot open < file.txt $!";
while (<$file>) {
    unless (/^A/) {

        # please note "$fh" filehandle to write to
        print $fh $_;
    }
}
close $file or die $!;
close $fh   or die $!;
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