You must have done something weird in my_sub because for me, warnings already tell me that I'm using a closed filehandle:

#!/usr/bin/perl

use strict;
use warnings;

open my $fh, '<', $0
    or die "I can't read myself";
open my $fh2, '<', $0
    or die "I can't read myself (2)";
my @lines = <$fh>;
close $fh;

sub my_sub {
    my( $this_fh, $line ) = @_;
    warn sprintf "At offset %d: %s", tell $this_fh, $line;
};

while (my $line = <$fh2>) {
    my @data = my_sub($fh, $line); # copy/paste error, should have pas
+sed $second_fh here

    # and of course a bunch of non-relevant stuff here
}
__END__
tell() on closed filehandle $fh at tmp.pl line 15.
At offset -1: #!/usr/bin/perl
tell() on closed filehandle $fh at tmp.pl line 15.
At offset -1:
...
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Maybe you can show us what you were doing in my_sub with the filehandle and where Perl didn't warn there?