Yay!

Here's what happened: SMF, the forum software, will resize any image that you upload to the server to the specified maximum image size (was 65x65 px, I changed it to 100x100 px). When it does that, the animation is lost because of the way SMF resizes it. There's nothing there can be done about that.
However, if you link to an external image that is animated, you can let the HTML handle the resizing. I changed the avatar setting to this, because it defaulted to having the external image uploaded (thus resized with lost animation) to the server.
In short: if you specify an URL to an external image in your profile, it should work (like mine does now). Don't be put off by the size the image might be before you click 'apply' in the bottom of the screen, that might take some sidebar scrolling

OR upload an animated GIF that is no bigger than 100x100 pixels.
I noticed the image gets a bit distorted, but at this size I hope that's not a problem.

--> go to your Profile (top navigation bar)
--> click 'Forum Profile Information' in the left navigation thingie
--> specify URL to image: