I would see two issues with using any birch species in a vivarium. First would be, is it possible to treat the wood well enough that you don't introduce any pathogens into the tank (even with heat treatment, there are many fungi and bacteria that will survive). Second, most of the birch species that grow around here are not very strong harwoods and breakdown quickly once they are dead. I would worry that the wood would decay even faster under the conditions that you would have in a viv.
One of the reasons that a lot of people like Cypress is that it grows in water, so is very impervious to most decay organisms. You might want to check into some of these species.