Facebook has started to roll out changes to profile page announced last week. One interesting new feature is the ability to set a looping video as your profile picture, not different from those GIF animations you see on the web.
To set it up, go to your profile page on Facebook app on iOS. Tap on your profile picture and select “Take a New Profile Video” or “Upload Video or Photo”. If you don’t see these options, then the rollout has yet to reach you.