How do you turn on comments for only the last page of a paginated post?

The easiest thing to do is to wrap the code that displays the comments in a conditional like this:

if ($page === $numpages) {
  echo 'last page'; // this is the last of the <!--nextpage--> pages
}

That requires a theme edit, so it is only a good solution if this is a theme you’ve constructed and maintain.

The variables (global) $page and $numpages are set by setup_postdata() and so should be reliable inside any properly constructed Loop.