Note edit form does not have a submit button

_flag_note_form(&$form_state, $action, $flag_name, $content_id, $flag_note = NULL) contains this check:

  if (is_numeric($content_id)) {
    $form['flag_note']['submit'] = array(
      '#type' => 'submit',
      '#value' => $flag->get_label('flag_short', $content_id),
    );

However, there is a problem with how this function is called:

ADD form:
    _flag_note_form($form_state, 'flag', $name, $form['#node']->nid, $note));
EDIT form:
    _flag_note_form($form_state, NULL, $flag->name, NULL, $flag_note);

With $content_id set to NULL, is_numeric will always evaluate false, and so there won't be a submit button.

Rather than adding that button back in, which places it inside the fieldset, I added a submit button next to the cancel button (outside the fieldset).

function flag_note_edit_form(&$form_state, $fnid) {
  $flag_note = current(flag_note_get_notes(array('fnid' => $fnid)));
  if (empty($flag_note)) {
    return FALSE;
  }
  $flag = flag_get_flag(NULL, $flag_note['fid']);
  $form = _flag_note_form($form_state, NULL, $flag->name, NULL, $flag_note);
  $form['fnid'] = array('#type' => 'value', '#value' => $flag_note['fnid']);
  $form['submit'] = array('#type' => 'submit', '#value' => t('Submit'));
  $form['cancel'] = array('#value' => l(t('Cancel'), $_GET['destination']));
  return $form;
}

This works as expected.