I recently download the Facebook app for iOS. It doesn't connect to the internet unless wi-fi is enabled. All other apps are able to connect to the internet with wi-fi disabled, just not Facebook. I am running iOS 9.3.
The only possible thing that I could think of is that you may have disabled maybe accidently Cellular Data for Facebook and to verify you could go into Settings -> Cellular and verify that under USE CELLULAR DATA FOR: is turned on for Facebook.
Another rare possibility could be that there might be a DNS issue with your cellular data provider whereas it wouldn't be able to reach Facebook on its own network.