Well, as I mentioned, in either case, with Facebook and/or Facebook Messenger, I haven't really noticed much of a battery difference when I have them running or not. Now, again, this kind of thing can be different for people of course, as many things are, but I'm sure there are many others in the same situation as me where having those apps in the background doesn't really seem to take much of a hit on their battery compared to not having them in the background (assuming iOS 7 background refresh is off for them).