I would be very reluctant to run a game via Facebook, I've played plenty of PBEM based games and chat-based (IRC style) games but I can't see Facebook as an ideal medium for this; I'll go through the reasons why for each style:
I'd recommend looking at This Question for more details about how Play By Mail games work.
- Post-size limitations: Facebook has upped it's maximum post size to 5000 characters, which sounds lots - but it's not. Any in depth PBEM style game can easily exceed this several times over in some cases.
- Character ID As mentioned by @DavidBall facebook doesn't like you creating character profiles for Facebook, you may well end up with deleted characters or indeed groups that you're using to store posts to contain the information if you're unlucky. This really won't help immersion if you're posting as "Bob Smith" rather than "Commander Jackson of the USS Apocalypse".
- Formatting Facebook really doesn't present a good medium for reading vast amounts of text, which is the core of the story based PBEM, the whole page is crammed into only a relatively small amount of screen and you have pretty limited options for changing font size, etc when reading it. Quite often different coloured text is used to indicate new parts of a message - facebook won't help you there.
- Got to sign up This is a biggie for some people, I know quite a few people who object (for many and varied reasons) to Facebook and refuse to sign up to it in any fashion.
Play By Post
Play by post is generally where players post in real-time about their game, mechanics (dice) may or may not be involved or it can be entirely freeform. I've mostly played the latter, I've generally found voice chat to be better for live (ie tabletop style) games.
Facebook is actually a bit better medium for play by post; but you do have the same problems as PBEM's and as well you have some other problems:
- Minimal support for utils Yes you can install apps for this, but you can't generally share things easily between people or roll dice for everyone if you need to.
- Distractions When you're playing a play by post you want people to be concentrating on the game, Facebook is going to be pretty bad for other people messaging them, post updates being lost or people just plain getting bored and playing FarmVille or something instead - yes people can switch tabs if they want to do this with other systems but having it available on the same tab as the one you're trying to play is really not a good idea.
I really can't recommend Facebook as a RPG play-by-post medium; it's not set up for it and there are lot of websites, systems or just email lists that are much more suitable for it.
If you're doing immediate style play-by-post find a website or use IRC.
If you're doing Play By Mail style use an email list.
At least that's my recommendation!