There are 298075 registered members of GBATemp as I am writing this. The Mr.Douchebag GIF animation you have in your signature is exactly 71787 bytes big - this is about 72 kilobytes, or KB for short. Now, if each and every member of GBATemp would have a GIF avatar like that, they would collectively take 72*298075 KB, which is 21461400 KB, which is 20958 MB, which is 20.46719 GB. Yes, over
20 gigabytes. These 20 gigabytes would be shuffled
over and over and over again across multiple devices users use, multiplying expotentially and weighing heavily on the bandwidth of the website without any particular use other than the user's fancy.
Now, unless GBATemp staff devises a method of storing avatars elsewhere, for example external hosting sites like Imgur, Minus, Photobucket and the likes, the chances of such an upgrade taking place are slim to none. Even if such a method
was devised, GBATemp would suffer as the pages would load sluggishly with thousands of animated GIF's loading from more or less reliable servers from all across the globe, devolving your browsing experience into a crawl. Of course this is assuming that
everyone would use an animated GIF as an avatar which clearly wouldn't be the case - a great deal of users don't even
have an avatar in the first place and on top of that many users are inactive, but what's a rule for all has to also be a rule for the individual - there are no exceptions, sorry.
I hope this post clarifies as to why we don't have animated avatars.