You can use animated images anywhere, Eastcoastmom! If you're using them on another forum or on Facebook, just make sure your image meets their maximum size requirements. Some forums only allow an avatar that's 100 by 100 or smaller so an avatar that works here might not work there.
I don't remember if you're able to use an animated image for your profile picture on Facebook. I'm currently using a non-animated photograph for my Facebook profile.