Oil has a very specific purpose in the engine, and that's to provide lubrication. The metal parts of an engine are constantly pressing against one another and creating friction. Without oil to provide a buffering layer in-between say, an engine's pistons and the chambers they move through, the engine would seize up and then fuse so that it would be nothing but a hunk of useless steel. However, not all oils are created equal, and the longer oil moves through an engine, the more impurities it picks up and the less viscous it becomes over time and miles traveled.
Changing out old oil for new oil improves gas mileage by a few cents per gallon, on average. The reason for this is fairly simple if you look at the physics. New oil is highly viscous, slick, and it allows the engine to practically glide with minimal friction. The older oil gets, the less viscous and the gummier it becomes. It still provides the necessary buffer, but there's more friction in the engine. The more friction there is, the more the engine has to work to overcome that friction. The harder the engine has to work, the more fuel it's burning. Since new oil requires the least amount of work, the engine is burning slightly less fuel.
The physics of friction are also what allows synthetic motor oil to provide better gas mileage than traditional motor oil. Engineered for a higher viscosity, synthetic motor oil creates a slicker surface on the engine than traditional motor oil does, and thus it saves even more energy. However, a word to the wise is necessary on synthetic oil. If you don't use it for the life of your engine, you may not end up getting the kind of savings you want. Synthetic motor oil is more expensive than traditional motor oil, and even though it does provide a fuel and maintenance savings, it has to be used for some time for those savings to be fully realized.
