I'm using the latest version of the Facebook SDK, version 3.1.2, for Unity. My problem is somewhat similar to this question. The differences being that my iOS builds work just fine. And my Android builds aren't getting null references. Instead, My android build will just not log in.
When I click my login button, it'll bring up the dialogue screen to give the app permission to access the basic stuff, public profile and friend list. Upon hitting the ok button, it'll return me to my app but fail the login. I found the AndroidFacebook class and noticed that OnLoginComplete function has a string message param where it should grab the user's id and access token from. So I put a debug message in there to see what was being returned. It turns out that although I press ok to give the app access to facebook, it's saying that the process was cancelled. {"cancelled":true} is the Json string being returned. I've doubled check my app settings in the facebook developer page and the Package name, class name and debug hash key all match what I have in the Unity Facebook settings. I've even tried with the Sandbox mode on and off.