How can I include a dropdown in menu_local_tasks tabs?
For the drop-down menu you need include drop-down JavaScript in the "page callback". See drupal_add_library() function.

And I recommend for you pay attention to "CTools module".

"Drop-Down Menu" Code from CTools module:

  // Show examples of ctools javascript widgets
  $output .= '&lt;h2&gt;' . t('CTools Javascript Widgets') . '&lt;/h2&gt;';

  // Create a drop down menu
  $links = array();
  $links[] = array(
    'title' =&gt; t('Link 1'),
    'href' =&gt; $_GET['q'],
  );
  $links[] = array(
    'title' =&gt; t('Link 2'),
    'href' =&gt; $_GET['q'],
  );
  $links[] = array(
    'title' =&gt; t('Link 3'),
    'href' =&gt; $_GET['q'],
  );

  $output .= '&lt;h3&gt;' . t('Drop Down Menu') . '&lt;/h3&gt;';
  $output .= theme('ctools_dropdown', array('title' =&gt; t('Click to Drop Down'), 'links' =&gt; $links));