Does weight affect the fuel burn calculations in a light GA airplane?
The engine doesn't care how heavy the airplane is (or how well it is flying), but you might.

As long as you are cruising around at a particular altitude and power setting, then the engine is going to consume fuel at the same rate.  A heavier airplane will require increased lift, which will induce greater drag, and will have a lower forward airspeed as a result of that same power (assuming it is not so heavy that it prevents you from maintaining altitude).

If on the other hand you have a particular airspeed you want to maintain, then the power needed for that speed will depend on weight.  Lower weight, less power necessary, less fuel flow required to supply the power.