Facebook is rolling out animated / video profile pics.
Your profile and facebook app need to be updated to
allow uploading a video for your profile image.
Lunapic will let you convert an animated gif to a
Video format (like mp4) to use in your facebook
video profile pic.
Just go to File -> Save and select MP4
(My phone didn't support saving this, so I had
to use a desktop computer to save video file and
transfer to my phone)
Once the video animated gif is on your phone, you
can use the Facebook video upload to set it as
your profile pic.