open (RECORD_NUMBER, "counter_data.dat") or die;
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With 2-argument open, not specifiying the open mode, you open the file for input only. Either open it for input, best done explicitly with 3-argument open, close it after reading, re-open it in write mode then - or open it for update. for die see below.

$count = <RECORD_NUMBER>;
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The line you just read might contain a line feed character at the end. Use chomp: chomp($count = <RECORD_NUMBER>);

$count = $count + 1;
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Better written as $count++; - see perlop.

print RECORD_NUMBER "$count";
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Useless variable interpolation within double quotes. print RECORD_NUMBER $count; is just fine. But then, you might want to append a "\n".

close (RECORD_NUMBER) or die;
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Big plus for checking the return value of close. It's rarely seen, but should be done on every close. You might say or die "Can't close filehandle RECORD_NUMBER: $!\n" - so you know what went wrong.

I would write your code probably as

open (RECORD_NUMBER, '+<', "counter_data.dat") or die;
chomp($count = <RECORD_NUMBER>);
$count++;
seek RECORD_NUMBER, 0, 0;
print RECORD_NUMBER $count,"\n";
close (RECORD_NUMBER) or die "Canīt close filehandle RECORD_NUMBER: $!
+\n";
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The seek call rewinds the file to the beginning. If I did not, the new number would just be written at the actual file position - after the line read in.

--shmem

_($_=" "x(1<<5)."?\n".q·/)Oo.  G°\        /
                              /\_¯/(q    /
----------------------------  \__(m.====·.(_("always off the crowd"))."·
");sub _{s./.($e="'Itrs `mnsgdq Gdbj O`qkdq")=~y/"-y/#-z/;$e.e && print}