In my opinion this has no influence at all. I would say the biggest influence is weather. i bow hunt almost every day and about the only time I don't see deer is if the weather is extremly bad. I would say the thing that influence deer the most is if where you are hunting gives deer a reason to be there.

First you have to know what a full moon is and what a new moon is.
A full moon means it is 180degrees opposite of the sun, so it is reflecting the most sunlight, and fully overhead around midnight. That means the entire night is lit.
A new moon is mostly invisible, because it is very close to the sun and has its fully shaded side facing the earth. That means the night is lit by stars only -there is no moon in the night sky to reflect the sunlight.
Animals that rely on some light, like crepuscular deer, very well might take advantage of a full moon to extend their feeding time, making their movements a bit more difficult to predict during that week.
So...what? Ask yourself - what phase is the moon in right now? Can you feel it? No? Then why do you think other mammals do? They know they can see better when the moon provides light. You would, too if you went out every night for a one-hour walk without a flashlight.
There's another post on this site attributing deep mystical significance to the cosmic alignment of the sun and moon, but I have to agree with hunter1849, weather conditions trump any celestial emminations.