One thing that has bothered me about facebook is that I cannot embed a chess game into a post. Sure you can put a link to a replayer like tuxtown or chesstempo but that would mean you have to click and go to a new page outside of Facebook.
The above is a rough way of embedding a full game via animated gif into a Facebook post. It immediately starts moving which is nice. However there are no controls like move forward and backwards, pause etc. I know it's possible to develope a Facebook app that can allow for this but so far no one has come out with one.
So please if there is a capable Facebook developer out there, please do the chess world a favour by making it! It will definitely make Facebook a more chess friendly place :)