A text file should end with a newline. Your code expects it, that's why you are using chomp.

Also, you should check open for success. Generally this is done with die:
open COUNTER, $counter_file or die "Cannot open $counter_file: $!";

Finally, you don't need the while block. If you know the count will always be on the second line:

my $counter_file = 'counter.txt';

open COUNTER, $counter_file or die "Cannot open $counter_file: $!";
    my $first_line     = <COUNTER>;
    chomp( my $counter = <COUNTER> );
close COUNTER;
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$counter++;
open COUNTER, '>', $counter_file or die "Cannot open $counter_file: $!
+";
    print COUNTER "$first_line$counter\n";
close COUNTER;
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Assuming you keep ignoring the warnings merlyn mentions above.