It took me a little while to sort this one out... but obviously it can be done.
You just have to use an off-site link to make things work, rather than the typical upload from your computer method.
Here's how :
In your profile for the avatar there are 3 options.
1) Upload from your computer
2) Upload from a URL
3) Link to off site avatar. (an image that is hosted off the site meaning PhotoBucket, Imageshack, etc...)
4) Enter the link to you animated Gif in the link to offsite avatar box.
5) Click save.