As pointed out, on the left operand of the or you have scalar context. List context propagates from the "for" onto the right operand of or. Now the catch: .. is two completely different operators in list vs. scalar context. In scalar context, it is the flipflop operator, and the constants 1 and 10 are implicitly compared to $. (the current readline input number.) In list context, it generates a list from the specified range. So what your code produces is something like:

my $flipflopcount;
print for do {
   my $flipflop;
   if (!$flipflopcount && $. == 1) {
      $flipflop = ++$flipflopcount;
   }
   if ($flipflopcount && $. == 10) {
      $flipflop = ++$flipflopcount . "E0";
      undef $flipflopcount;
   }
   if ($flipflop) {
      $flipflop;
   } else {
      (1.11);
   }
}
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Try this: perl -wne'print for (1..10 or 1..11)' foo where foo is a file with 12 or so lines. You will get "12345678910E0" as the output for the first 10 lines and "1234567891011" as the output for each line thereafter.

To the monk confused by the difference between scalar(1..10) and scalar 1..10, that's a precedence thing. The latter is equivalent to scalar(1)..10 so the scalar() is not forcing .. to have scalar context.