Depends on the burl--in some cases down to the specific piece. If there is no spalting, and it's a hard species to start with, probably going to be good. I've never had to stabilize thuya, coolibah, any mallee, for that matter almost any Australian burl (thuya is Moroccan). The various eucalyptuses are usually pretty firm as well. On the other hand, redwood burls in my mind are too soft for a pen without stabilizing--though i've been pleased with bowls that were not stabilized as they don't take the abuse a pen takes. The rougher the growing conditions, the harder/denser the wood/burl is likely to be (unscientific opinion).
In between--domestic maples and cherry burls i rarely stabilize, but often run into voids so i keep a supply of powder stones (turquoise, malachite, lapis), metals (aluminum, copper, brass), wood dust, tea/coffee grounds and use either epoxy or super thin CA as a binder...depending on the size of the void, colors, how deep i think the glue needs to carry, etc.
I'm your basic health insurance guy who dabbles--so even though the above sounds like experienced or educated expert--i'm a hack who lost some pieces and saved some pieces trying stuff i read about here from people who really know what they're doing. Don't stop reading, asking and experimenting!!
Good luck!!
earl