Well here in Idaho where I hunt, primarily archery hunts, moon phases do have an effect on their daytime movements. When you go out early September, the moon is usually high and bright at night. You will hear all sorts of elk calls all night long. Then as soon as the sun begins to rise and day breaks they get very very quiet. They are there and still move, but talk much much less. This can make it very difficult to find them if your not already on top of where they are.
The third week of September here is when the moon begins to head into the new moon phase. As the week progresses, the elk will talk longer into the morning hours, and really helps to locate them early morning. The rut itself also tends to follow this part of the month and their rut drive is increasing. Hunting during the peak of the rut has proven time and time again to produce the best advantages of calling and attracting the elk into range.
Does moon phase have an effect? I think yes, as it helps to time the rut peaks here where I hunt. Historically reference to the elks rut, it follows the moon phases exactly.