Update cart on checkout page with tax rule
There is no way right now to show the update without custom code, though it could be done by adding AJAX functionality to the form like the shipping module does. You can see how it updates shipping rates when address data is entered, and if I'm not mistaken it even provides a hook for you to add additional AJAX commands to the response. You should be able to update the order at that time and update the cart contents checkout pane display, too.