The old theory that deer feed all night and sleep all day during the full moon is just that, a theory. The theory goes that they do this because they can see better from the full moon's light, hence the reason they feed all night.
The truth is, deer don't need the light of the full moon to see. They can see just fine without it. Deer naturally feed and move about more at night because it's cooler and, in most cases safer. They see very well at night, with or without the light from a full moon. Actually, I believe deer movement increases during the full moon period. But, like as mentioned, it depends on a lot of variables(temperature, cold fronts, barometric pressure, stage of the rut, etc.).