It is my understanding that if you're applying a CA finish, that's effectively the same thing as stabilizing the wood. So if that's your plan, then you only really need to stabilize if the wood would otherwise fly apart during turning. I've turned a fair amount of un-stabilized buckeye burl, for example. That stuff is like styrofoam when you turn it, but when it's CA finished, it's just as smooth, glossy, and strong as anything else.
If you don't want to use a CA finish, you'd probably want to stabilize more or less any burl.