In simplistic terms, the energy required to overcome a load must come from the fuel. RPM being equal, a loaded engine requires more energy than an unloaded one, to run at the same RPM.
For a relatable physical experience, consider cycling on the flat versus cycling up a hill, where your speed (wheel rpm) is the same- cycling up a hill requires more energy (more pedalling effort) and you'll tire (deplete your energy reserves) sooner. If you cycled slower and only put the same amount of pedalling effort in as cycling on the flat then after X amount of time you'd have used the same amount of energy but you wouldn't have travelled as far. To make the travel distance the same you'd have to burn more fuel
So in short, two engines running at 3600 rpm, one loaded one not, if they both started out with the same amount of gas the loaded engine will run out of gas sooner because its rate of fuel consumption will be higher
That's the question in the title dealt with. You've asked a few different ones though:
If I ran the engine at full throttle without any external load connected to the output shaft, would it consume the same amount of fuel if I ran the engine while connected to wheels on the ground?
It depends on whether there's some mechanism that means throttle position sets the RPM or whether it's just a basic air flow control. If there is no particular RPM being strived for, then the engine will run as fast as it can for that throttle opening for that load. When unloaded it will run faster than when loaded, and it will consume fuel at the same rate in either case. Some of the energy in the fuel is used to overcome the load, the engine runs more slowly when loaded but it still requires the same amount of input energy to establish this new steady output state with a load that is taking some of the energy.
If however, there is a mechanism that senses the drop in RPM and opens the throttle more to compensate, then more fuel is consumed to maintain the steady state of the RPM, and the extra fuel provides the energy to overcome the load
It's probably also worth mentioning in this explanation is that all engines have a power curve; peaks where their efficiency (the amount of energy they can usefully extract from fuel) is greatest. Most engines reach a point where running faster consumes more fuel without delivering a useful proportional increase in the amount of torque/power they develop. In one of these zones it may be possible to achieve a set of conditions where the throttle position is the same, the unloaded engine is running at a higher RPM but suffering a disproportional loss of power development and applying a load that brought the RPM back down to the engine's most efficient power developing rate could see the same fuel consumption rate, same throttle setting, slower engine speed - essentially trading the energy used to overcome internal inefficiencies at the higher rpm for the ability to overcome the useful load at the lower rpm - you sort of get the work on the load "done for free" as compared to running a higher engine rpm but losing efficiency. It's a narrow usable application case, but it perhaps highlights the sense in running an engine so it is operating at peak torque or peak power (the two don't necessarily coincide) depending on the application; that way you're getting the most work done for your money spent on fuel, if the load on the engine is sustained and heavy enough to require all the engine's useful output effort. (If you're mowing a well tended lawn or other light weight task far beneath the capacity of a powerful mower, then some low engine speed gets you more for your money than running at peak output because you're not supplying a load that can usefully consume all the engine's output)
In both cases, the engine would run at the same RPM until the fuel runs out. Which case would run out of fuel sooner, or would it be the same time?
This is a paraphrase of the title question. The engine working harder to stay running at the same speed will consume more fuel to do so because of the principle of conservation of energy in a system - energy in equals energy out. Applying a load demands more energy out, so more energy must go in. In practical terms this means supplying more fuel (air gasoline mix) via a wider throttle opening
EDIT: This engine has a governor set to 3600 RPM, which I am assuming would limit the top RPM, thus preventing an RPM above the redline?
Now there is the mechanism talked about earlier, where instead of setting a throttle opening and having the engine speed up or slow down as the load changes, the governing mechanism on things like mower engines works more along the lines of the throttle lever being used to select the desired rpm. If a load is applied and the engine slows down, a mechanism opens the throttle more to counteract the imposed load and strive to get back to the set RPM, using more fuel in the process. It's a reasonable bet that your engine will run a good amount faster than 3600 rpm, but it may be set up that thats the maximum rpm you can request because it's the peak output of the engine and there isn't much point making the carburettor more sophisticated to cope with the changing demands of higher rpm running, especially if the engine design isn't optimised for it either
