I don't think so. The file is the exactly the same as was used previously.
Avatars are deleted from the server once changed. It doesn't matter if it's the same file.
I remember the avatar-related security issues in the past. But I'm not aware that these were caused by gif animations.
I'm unsure if it is the same security issue that the forum had, although it seems like it could be the case, you can read about how an attacker can run arbitrary PHP code through gif images
here. As said, I'm unsure, however considering that the security exploit was executed by
running arbitrary PHP code it could be possible that the two are linked.
Note that my animated avatar used less than 1% of the maximum allowed file size. If I'd be mean, I could upload a new 100 KB static avatar just to express my frustration by draining more forum resources...
I don't think that storage space is the problem with your avatar.