Changing the avatar already triggers a rebake of posts containing quotes, but it’s currently quite resource intensive to find and rebake all posts. So, it’s possible that some of them won’t be updated. Post revisions aren’t updated at all.
I have plans to fix all of this when I’m working on improvements for user renaming/anonymizing in one of the next releases. It will require a new table that keeps track of all the places where a user is quoted, mentioned, oneboxed, and what not.
If you want to work on this, please keep in mind that rebaking just to update the avatar isn’t needed. I’d suggest reusing code from UpdatUsername. It’s a lot faster.