What could I do?
I changed line 292 in rate.module from:
drupal_add_js(drupal_get_path('module', 'rate') . '/rate.js', array('scope' => 'footer'));
drupal_add_js(drupal_get_path('module', 'rate') . '/rate.js', array('scope' => 'footer', 'group' => JS_DEFAULT, 'weight' => 200));
Just for testing, but did not help. The I commented the line out and added the line
scripts = rate.js
to rate.info. Now it works again. I know that this is not a good solution, because now the JS is loaded every time on every page. But I am a bit helpless at that point not having any idea what could be the cause of that strange issue.
An other problem is, that rate widget refuses to work with ajax, if it is loaded via ajax it self.