E. Linacre and B. Geerts
A century of daily rainfall data from Sydney shows no connection with the phase of the Moon, apart from very slightly fewer heavy falls at the time of the full moon (1). Similar results have been found elsewhere. Data from Jakarta on monsoonal rainfalls (i.e. over 38 mm/day in December - March during 1864-1945), and from Magalore on the west coast of India (1901-50), show a tendency for more rain at certain times of the lunar cycle, especially in years with relatively few sunspots (2). At Jakarta, most rain occurs shortly before full moon, and least just after new moon. The deviation is 25% from the usual in years of below the median number of sunspots. At Magalore, most rain occurs when the moon is in the first quarter, and least in the last quarter, but the amplitude of this cycle is smaller.
Nighttime minimum temperatures on Earth are a little higher under a full moon, compared to a new moon. This has been demonstrated by means of an analysis of global surface temperature data. The difference is only about 0.01 K, and this difference is consistent with the extra energy the nocturnal Earth receives under a full moon. The difference clearly is insignificant, and in most situations it can't be measured. Weather stations measure the temperature with an accuracy of at most 0.1 K.
The moon exerts a gravitational pull on the atmosphere, but the resulting vertical air motion is negligible, even during full or new moons. Along coasts with large tidal differences and broad beaches, the phase of the moon can have a small effect on the local wind circulation, either through thermal effects (i.e. the strength of the sea breeze) or mechanical effects (e.g. onshore flow forced by the upheaval of water). As such, the phase of the moon can indirectly affect temperature, cloudiness and precipitation in coastal areas. Lack of evidence suggests that this effect is small as well.