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.
More weight means the plane will need more lift, and more lift implies more drag. For example, and for a constant airspeed, an increment of 10% in weight will mean 10% more lift, achieved by increasing the AoA, and that will mean 10% more drag too. The power required will be 10% higher also, implying 10% more fuel consumption...
For heavier loads, the relation would not be so (approximately) linear, because the airspeed will be probably lower, the AoA higher, and the induced drag far higher. Thus, the power required (and the fuel consumption) will rise disproportionately.
Short answer is: you would burn more fuel.
1 You need more energy to counteract gravity during the climb.
2 At a given speed, a higher mass requires more lift, and more lift implies more drag, more drag requires a higher power setting to achieve the same speed, and a higher power setting means a higher fuel flow.
Your POH (part 5) might explain how to calculate the new required fuel.
In short, yes. However the limited flight envelope and useful loads for these kinds of airplanes combined with limited increase of induced drag will only decrease speed a few knots or so between the airplanes empty weight and maximum takeoff weights. This only translates to a couple of extra gallons in fuel savings.
To compensate for this, the OEM approved performance charts and flight planning literature in the AFM are usually done at the maximum takeoff weight of the airplane for conservative purposes; the airplane may perform slightly better in optimum conditions but manufacturers err on the safe side for the average private pilot.
