Force authors to Preview a post before publishing

This is completely untested but I think the concept is good, and good idea too though I’d have an angry mob with pitchforks to face.

1. Set a marker for your post when the post is previewed. You can save
   it in either $wpdb->options or $wpdb->postmeta. I haven’t
   quite decided which would be better, and it probably depends on
   details of the implementation. (But probably $wpdb->postmeta).
2. Check that marker before you allow a post to be published.

So…

function set_preview_marker_wpse_99671() {
  if (is_preview()) {
    // set the marker
  }
}
add_action('pre_get_posts','set_preview_marker_wpse_99671');

Then check the key before you allow publishing.

// code shamelessly stolen from another answer
// http://wordpress.stackexchange.com/questions/96762/how-to-make-a-meta-box-field-a-requirement/96767#96767
function req_meta_wpse_96762($data){
  $allow_pending = false;
  if (isset($_POST['meta'])) {
    foreach ($_POST['meta'] as $v) {
      if ('your_required_key' === $v['key'] && !empty($v['value'])) {
        $allow_pending = true;
      }
    }
  }
  if (false === $allow_pending) {
    $data['post_status'] = 'draft';
  }
  return $data;
}
add_action('wp_insert_post_data','req_meta_wpse_96762');

You might want to unset that key when the post is published to force the preview check again if it is edited.