That ole' moon is circling around the earth every day, whether full moon or new moon or in between. Last I heard, it exerts the same gravity, regardless of how much sunlight is shining on it.
For most plants, energy production is maximal during the day (sunlight), while energy utilization for growth occurs mostly at night, with little impact from the smattering of photons from moonlight.
The moon is such a conspicuous and changing object in the night sky, it has been tempting throughout history to assign astrological, numerological and agronomic importance to its appearance.
Oceanic tides at shorelines require open expanses of water to accumulate enough of a gravitational mounding to be at all visible. Examine the "tides" of a large (say, 2 acre) pond. Nothing. I would guess that the lunar gravitational effect on plant shoots and roots is several orders of magnitude smaller than the effect of immediate conditions (weather, herbivore predation, nutrient ratios, soil compaction).
Although I know of no ill effect of planting by the moon phases, I wouldn't expect dramatic benefits.
Bob