The pic won't be animated in your iPhone camera roll, but if you share it or send it anywhere that supports the gif format, it will be animated.
If you have already created a profile pic and did not save it during the time that you created it, then you cannot go back and save it later.
For iPhone, make sure PHHHOTO has access to your camera roll:
- go to your device SETTINGS
- tap PRIVACY > PHOTOS and enable PHHHOTO