The short answer....Once you uploaded it, it's not a gif anymore, so it won't animate.
The long (geeky) answer why. When the avatar images are uploaded they are manipulated in code and stored as JPEGs no matter what the original format is. The data is stored as an image type in the database. Then code renders the image on your posts. This is why if you right click most of the user's avatars and choose properties you see that the image file path is .../users/avatar.aspx?userid={%member_user_id%}. "avatar.aspx" is simply a server code page that retrieves the image based on the request (the ?userid= part) and renders it.
In the case of animated gifs I don't know how it decides which frame of animation to choose as the part to convert to JPEG. Probably the first one...(but since gif ani's get optimised and can be complex I don't know for certain).
We do have remote avatars enabled, so if you have web hosting that allows offsite linking, you may be able to specify the image as a gif from your website / hosting account (should you have one). I haven't tried this so I don't know if it works (IOW that the forum still doesn't manipulate the image even though it's being retrieved remotely)...