As per this The Gravitational Force of the Earth and Moon on Each Other Moon is pulling us with a force equal to 1/300,000th of our weight. I have read this Is the fuel burn for a satellite launch affected by the position of the moon relative to the launch site? NASA do no wait for moon to come overhead of rocket.(why?)
Considering this heavy rocket http://www.spacex.com/falcon-heavy have mass of 1394000 kg .
If the rocket is launched when moon is overhead it can save upto 13661200/300000 = 4.64 kg of load or might be more? (as we go above in space the gravity of moon increases and earth gravity decrease.)
Update : removed W = 1394000 * 9.8 = 13661200 kg as it was in 1g environment.
How to calculate pulling force to rocket when below planets are in line?(diagram just for reference)