[EDIT] *Provided better replacements.*

Try this:

In *template.php*, replace:

    echo '<div id="vote_counter">'. $votes.'</div><br>';

..with:

    echo '<div id="vote_counter-' . $post->ID . '" class="vote_counter">'. $votes.'</div><br>';

In *voteme.js*, replace:

    jQuery("#vote_counter").html(response.vote_count);

..with:

    jQuery("#vote_counter-" + post_id).html(response.vote_count);

You should also declare `post_id` and `nonce` as local variables; i.e.:

    var post_id = $(this).attr("data-post_id"),
      nonce = $(this).attr("data-nonce");

Hope that helps.