I wonder why most of this thread is talking about file handle safety and none of those even mention what will happen if a line in the file is rm -rf /. (thanks for the posts that did mention "do not use eval". eval is dangerous!)

I would not even consider testing this script without a minimal validation of it's input. A start could be:

use 5.16.3;
use warnings;
use autodie;
open my $in,  "<", "numbers.txt";
open my $out, ">", "sum.txt";
while (<$in>) {
    m{\S}    or  next; # skip empty lines
    m{^\s*#} and next; # skip comment lines
    m{^[-.0-9+*/ ]+$} or die "Line too dangerous to evaluate!"; # . fo
+r floats
    # pattern needs extending for math (sqrt, log, sin, cos, ...) supp
+ort and exponentials with e notation
    # and 'x' for hexadecimal, '(', and ')' for grouping etc etc. Anyw
+ay, it is dangerous.
    # I'd not use eval for safety reasons, but if your validator is st
+rict enough
    my $result = eval $_;
    $@ and die "Invalid expression on line $.";
    print $out $result;
    }
close $in;
close $out;
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