No. You can log into FB app. You can remain logged in and close it.
It does nothing until you open it.
By being logged in, EMAILS for facebook, get intercepted by the app, and you get the notification. The app doesnt need to be running in the background to do this.
FB app will only load the notification, once you click on, at which point it starts retriveing and refreshing feeds, messages, friends list etc.
If you've closed it, but still logged in, it does NOT use data.
"That's it then. Cancel the kitchen scraps for lepers and orphans, no more merciful beheadings... and call off Christmas."