There are several problems with your code, chief among them is that you are not checking any of the IO calls. I altered it to:

use strict;
use warnings;

$|=1;
open(STDOUT, '+<') or die "open: $!";

print STDOUT "11111111111111111111111111111111\n" or die "print 1: $!"
+;

for (0..31){
  seek(STDOUT, $_, 0) or die "seek: $!";
  print STDOUT '0' or die "print 0: $!";
  sleep(1);
}
close(STDOUT);
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This fails with open: No such file or directory.
Although you can open an existing file handle as another, as others have said, it still won't work. Changing the open:

open(STDOUT, '+<', "&STDOUT") or die "open: $!";
[download]

now fails on the seek: seek: Invalid seek because it is not a seekable device.