I don't think I have that same problem. I opened FB and let it sit and my device went to sleep. I checked my battery usage and FB only uses a small percentage (as opposed to Display, Voice calls, Cell standby, etc).
I haven't have any issue with my facebook app. You should uninstall and reinstall the app, or clear Data and Cache in menu _ settings _ applications _ manage application _ Facebook. Also, tap uninstall updates as well.
Uninstalling noLED seems to have fixed the battery drain caused by the Facebook app. Weird that noLED would cause that, since I only turned it on once to see what how it would handle the notification. After that I turned it off but didn't uninstall it.
I had this exact same problem! I would check to see what was draining my battery, and Facebook was draining it the most. Like you, it recently started happening. I ended up uninstalling and reinstalling Facebook, as well as turning auto-refresh off.