The facebook API has limitations. This page shows what the service connect function in Mathematica can do:
http://reference.wolfram.com/language/ref/service/Facebook.html
fb = ServiceConnect["Facebook"]
fb["WallPostStatistics"]
This gives a number of statistics including "TotalLikes". This is only for more recent posts which are given by the API though.
You can also use this to find likes on pages and information about specific posts.
