Many moons ago, our profile pics could be animated gifs. Then the gifs became still shots.
Animated gifs are an abomination and a crime against all that is good and holy ... unless they are on Imgur in which case they are great.
The reason for no animated avatars is that they are aesthetically distracting, harder to police and gifs really suck battery power and bandwidth if you're on a mobile device. They are also harder to resize compared to a static image which is an issue when a site loads images at different sizes for different screens / device resolutions.