What effect does weight have on the range of an airplane?

Weight will reduce the range of your airplane. Added weight will require more lift. More lift means more induced drag. More drag means more power is necessary and consequently greater fuel consumption. In the same airplane loaded with more weight at the same power setting you will arrive at your destination later (given the same atmospheric conditions). In order to increase range it would be wise to load the airplane with the C of G more aft (but still within limits).