It is not possible to remove an avatar from displaying but it is possible to change it. There are a couple of options I might suggest.
The first is to use a transparent image, making it look like you have no avatar. There is a transparent image hosted by yuku that you can copy, upload and apply. It is only 5px square, so not so easy to right-click and save. >< (between the arrows)
My second suggestion is to use the original balloon-head. It can be found here: