The solution is very simple. In template.php file implement MY_THEME_preprocess_node(). Example:

function MY_THEME_preprocess_node(&$variables) {
  //  dpm($variables);
  if($variables['type'] == 'MY_CONTENT_TYPE') {
    $date_field_value = $variables['content']['field_DATE_FIEL_NAME'][0]['value']; // find the correct value of your field

    $converted_date = strtotime($date_field_value);

    if(time() > $converted_date) {
      hide($variables['content']['field_DATE_FIEL_NAME']);
    }
  }
}

The solution is very simple. In template.php file implement MY_THEME_preprocess_node(). Example:

function MY_THEME_preprocess_node(&$variables) {
  //  dpm($variables);
  if($variables['type'] == 'MY_CONTENT_TYPE') {
    $date_field_value = $variables['content']['field_DATE_FIEL_NAME'][0]['value']; // find the correct value of your field

    $converted_date = strtotime($date_field_value);

    if(time() > $converted_date) {
      hide($variables['content']['field_DATE_FIEL_NAME']);
    }
  }
}

you can also use hook_node_view:

function MY_MODULE_node_view($node, $view_mode, $langcode) {
  if($node->type == 'MY_CONTENT_TYPE') {
    $date_field_value = $node->content['field_DATE_FIEL_NAME'][0]['value'];

    $converted_date = strtotime($date_field_value);

    if(time() > $converted_date) {
      hide($node->content['field_DATE_FIEL_NAME']);
    }
  }
}