Correct, I try to upload an animated avatar and it hangs for a few minutes then errors out.
It's been a few years since I've had to deal with it, but there was a security feature that could be enabled on a VPS that sometimes caused issues. But honestly it's been so long (and I'm getting old) I can't remember exactly what it was.
Are you sure you are using the latest version of ImageMagick?
Is this shared hosting?
One recommendation I found was increasing the PHP timeout limit. If you are on a shared hosting plan or an over-sold VPS you could have issues as PHP/ImageMagick has to process the image and if there are not adequate resources it can cause longer time to process.
Albeit an older version, but this may still work (I have all my XF sites offline currently)
I tried to upload a profile picture while I had ImageMagick PECL Extension enabled. This didn't work, nothing happened. When I switched to PHP GD it worked as it should. I switched to Imagemagick a while ago because PHP GD couldn't upload large attachments (the thumbnail didn't show).
xenforo.com
Have you tried uploading the animated avatar into a thread and see if that works? If not, very likely a timeout issue.