The most likes on a Facebook item in 24 hours is 3,868,253, as featured on the official page for Barack Obama (USA) in response to a post made on 07 November 2012.

The post, “Four more years.” and a picture of Obama embracing wife Michelle, was an announcement of his re-election as President of the United States.
http://www.facebook.com/photo.php?fbid=10151255420886749&set=a.53081056748.66806.6815841748&type=1