I agree with Joel that "Burn up usually refers to items that are consumed by fire; wood, paper,etc. "
That is the literal definition. There are also metaphorical uses such as "he is burning up with a fever". Of course, he is not literally on fire but his body temperature is hotter than normal.
Ben is also right that "burn up" and "burn out" are not interchangeable. As he wrote, "The primary meaning of "burn out" is "cease to function as a result of exhaustion or excessive heat or friction". The best way to understand "burn out" is to think of a motor that "burns out" because of excessive speed or lack of lubrication. Also, a light bulb can "burn out". The filament burns instead of glowing and the bulb ceases to function.
Hope that helps.