The oil is kept from being burnt by the rings around the cylinders, and the smooth walls of the cylinders' chambers. When you start to burn oil, oil is getting through. Normally this means new rings, at a minimum, but if the engine is particularly worn, it would need to be "re-bored" - a process where the cylinder chamber is drilled again, a fraction of a millimeter/inch wider, creating a smooth wall again, and then with slightly larger rings to ensure that the seal is kept.
