I’m actually surprised nobody has posted anything about this online, but it’s more or less possible to write an IMAP proxy for Facebook using their API. Sure, you could not write a full-fledge IMAP implementation, but you could get it to perform the most basic tasks, like reading messages (and therefore also store them locally).
I got the idea of making an IMAP proxy (or bridge, parser or whatever you want to call it) back in in August when Facebook announced new API calls to access the mailbox. Upon reading them, I realized that it could be used for a lot new things, such as this. Since then, I’ve been waiting for someone take on the challenge. I’ve been keeping an eye open for a projects in this area, but haven’t seen any. Therefore, I thought I might at least raise some interest and perhaps someone will take on the challenge.