Make no mistake: this is user-hostile. Facebook is actively creating channels to continue refreshing their app in the background when the user has explicitly stated that they do not want it to. Ironically, the best way to reduce the battery and data consumption of the Facebook app in the background is to switch Background App Refresh back on. Better still, remove the Facebook app from your phone, and perhaps replace it with Paper (US store only).
Every time I take a look at a friend's iPhone, Facebook is the app with the highest amount of battery usage in the background – even with Background App Refresh turned off. This has been going on for years, and instead of fixing the issue, it does seem like Facebook is always coming up with new ways to circumvent user control and consume more energy.
The fact that a company the size of Facebook can't optimize energy consumption of their iOS app is simply ridiculous. If they can but don't want to (because of processes they want to run in the background, constant notifications, etc.) – well, that's even worse.
My solution has been the same for the past couple of years: never install the Facebook app, and always access Facebook from Safari.
Update: With iOS 9's improved energy consumption stats, it's easier to guess one of the various tricks Facebook may be employing to stay active in the background and drain battery. On my girlfriend's iPhone, for instance, iOS 9 reports 5 hours of on-screen usage for the last 7 days, and another 11 hours of background audio usage with Background App Refresh turned off.
My guess is that Facebook is hijacking audio sessions on iOS by keeping silent audio in the background whenever a video plays in the app. And because, by default, videos on Facebook auto-play on both Wi-Fi and Cellular and few people ever bother to turn it off, that means there's a high chance the Facebook app will always find a way to play a video, keep audio in the background, and consume energy to perform background tasks. I'm not alone in noticing the mysterious "Facebook audio" background consumption, and video auto-play seems to me the most likely explanation at this point. I don't know if turning off auto-play may fix the problem, but I'd recommend doing that anyway to save data.
I wonder if Apple should consider additional battery controls to take action against shady practices like invisible background audio. What Facebook is doing shows a deep lack of respect for iOS users. I continue to recommend using Safari instead.
Update 10/15: In a statement provided to TechCrunch, Facebook confirmed they're aware of an issue causing battery draining for users, and they're working on a "fix". Facebook didn't provide any additional details on the nature of the issue.