This thread has been going on for some time now, and I have just stumbled upon it. It's the only place where I can see this issue is being discussed. I've made some observations that I'd like to share, because I too, am very frustrated with it. I have an iPhone SE running IOS 10.3.1. Facebook Messenger completely ignores WiFi. It doesn't matter if you are connected to WiFi, Messenger chews up your data plan at rapid rates. On my phone, it's about half a Gig every 30 minutes, WiFi or not. WiFi assist is off. But you all knew that already. So, I turned off cellular data access and background refresh in the app section of Settings->Messenger. NO difference, those levers are IGNORED by the Messenger application. I then turned off data usage in a second place under Settings->Cellular->Messenger. NO difference. Those levers are also IGNORED by the Messenger app. I then turned on airplane mode, then activated WiFi...it worked! No data plan usage. Why? My guess is that airplane mode is a federally mandated convention that MUST be developed into every mobile app. I also think Facebook is in an agreement with the major carriers to chew up data so we buy more and more....OR they would have fixed it by now.
So, if you want to use Facebook Messenger over WiFi, and not have your data chewed up at rapid rates, turn on airplane mode, then WiFi.
BUT, there is more! Due to Messenger misbehaving, I ran out of carrier data. My data plan had ZERO bytes left, and I was on what is called "Safe Mode" (which is a pathetic amount of 1x bandwidth I purchased for $5 until I get a data re-fresh). Without a data plan at all, and no WiFi, I got a Messenger video call!! The picture and voice quality were great! So in conclusion: 1. if you have data and want to keep it, use airplane mode when on Messenger Video BUT Messenger Video works just fine WITHOUT ANY DATA AT ALL!! I can't explain why, but can only refer back to my comment on a agreement with the carriers. Apparently Messenger is NOT an internet application, but rather communicates over a cellular network (not a WiFi internet type of network). That explains why Messenger calls appear in your recent "voice" call list, if you've never noticed, that is also strange.....