What would cause a car to lose large amounts of oil and coolant at the same time?

Cars do not burn coolant per say as cars burn gas. What makes your car use or go through a lot of coolant is if you have a leak someplace in the hoses, radiator system, radiator cap, freeze out plug, or too much straight water will make it run hotter.

Is this car heating up? Are you adding straight water every time? It also could be a blown head gasket, but then it would be so hot the engine would stop and seize up. Can you smell it? If it smells like antifreeze but you don't see any tell tail signs of water leaking, then I would first check all water hoses for a pin hole. Look over your radiator real good and use a good bright flash light even in the daylight, you will be surprised what you can see. look for water stains (colored) green or white or even rust color. This would be a leak or a old one.