Does weight affect the fuel burn calculations in a light GA airplane? When calculating fuel burn in say a Piper Archer, I usually just use the conservative estimate of 10GPH, but, does the weight of the aircraft affect my fuel burn? 

Assuming I have full fuel and am taking off a max takeoff weight, will I still burn the same amount of fuel in 4 hours as I would if I took off with only myself in the plane and full fuel?
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.