Because hay is its own form of grass or legume, unless you're looking to harvest hay, do not place it on your fledgling lawn. Its seed heads will come loose, sink into the soil and germinate. While alfalfa might be a wonderful crop in a field, on your lawn it's only a weed that will compete with your grass seed for nutrients and moisture. Weeds multiply and kill plants in their way.
Another problem with hay is that while it certainly will keep moisture on your seed, once the seed starts to grow, too much moisture can be a bad thing. Too much water will rot the roots of your grass seed. Additionally, as the hay decomposes, the microbes feeding on it can drain your lawn of needed nitrogen. Damp conditions and nutrient deprivation will kill your seeds.
Chances are what you think is hay on the better sowed neighborhood lawns isn't hay at all. Closely resembling hay, straw is actually a good covering for new seedlings. Because straw comes from the hollow stalks of grain plants, such as barley, oat, wheat and rye, there is no seed to take root on your lawn. You get all the grass seed protection and benefit without the dangers.
Use straw, as a mulch, once your seed is planted. Be careful not to spread the straw too thickly as the seeds need moisture and sun to thrive. Once the seed begins to grow through the layer of straw, gently remove the straw with a pitchfork. If you allow the straw to degrade on its own, you'll want to add nitrogen to the soil to offset the negative effects of the decomposition.