Does covering potatoes make them sprout quicker?

No. Potatoes are asexual. Meaning they can naturally reproduce, given the right conditions of time and light. Ideally, store potatoes covered, in a cool (not cold) and dark location. For example, under a cupboard or in a pantry. Even under the best of storage conditions, potatoes will dehydrate and start to sprout, so be sure to use the potatoes while they are still firm and fresh.