Playing devil's advocate here--
So why exactly are you dismissing it out of hand? Just because "phase of moon" sounds like witchcraft?

There is a distinct difference in wood cut at different times of year -- winter-cut wood has less sap; that's why we can get maple syrup in the spring -- the sap rises.

Now I don't know about any monthly cycles, but I'm not a tree expert. But it could be -- women have a monthly cycle; due to hormones rather than the phase of the moon, but it can correspond. Are you enough of a tree expert to say nothing of the sort is happening with trees?

Professional woodcutters in Europe often talk about the necessity of cutting wood at the right time, and sometimes use language that sounds downright medieval. I'm not going to make fun of them for it; they are the experts. I listen with some skepticism, and suspect that the words used are a shorthand for what is really happening (as in a woman's menstrual cycle) but until I see a scientific study I will give them the benefit of the doubt.