Ranking items (groups, nodes) by the number of Facebook likes
This question recently came up at work (small world), my thoughts:


Render the Facebook like button using the XFBML version of the button
Attach a handler to the click event (see the docs)



FB.Event.subscribe('edge.create',
  function(response) {
    // do some AJAX stuff here
    alert('You liked the URL: ' + response);
  }
);



In the above callback, fire an AJAX request back to drupal, to update either a custom table, or alternatively you can use the core statistics table. If you wish to use the core statistics table (or even create your own table) you can leverage the code from my module https://drupal.org/project/statistics_ajax

One advantage of using core statistics, is that it has views integration out of the box, although in saying that writing custom views data sources for your table is easy too.