When using a free power turbine turboshaft, does fuel burn get affected by N2 loading? I am asking as I want to in the future bench test a turboshaft for fuel burn, but I don't know if the loading would affect the fuel burn.
The free turbine shaft can be called $N_2$, but $N_f$ is more common.
Your fuel control for a turboshaft is not so simple. It requires a Fuel Control Unit and an engine control that needs to know how much torque is being produced at the $N_f$.
A fuel control unit [acts] as an intermediary between the operator's controls and the fuel valve. The operator has a power lever which only controls the engine's potential, not the actual fuel flow. The fuel control unit acts as a computer to determine the amount of fuel flow needed to deliver the power requested by the operator.
[A] turboprop or turboshaft engine control system has an additional job to do that is not shared by its turbojet and turbofan counterparts. It must control the speed of the propeller or the free turbine, and it usually governs the pitch angle of the propeller blades. (Source)
If you are not bypassing the FCU and other engine control systems, you do need a varying load to have control over the fuel flow (indirectly).
Running an unloaded turboshaft at high power is very dangerous, as the free turbine will be rotating at RPM's way above its design limits. Unloaded free turbines may also cause an auto shutdown.
Free turbine engines are now commonly fitted with a device to shut off the fuel supply at the HP cock if torque in the turbine output shaft suddenly falls to zero.
A drawback to the simple free turbine turboprop is its behaviour if the load suddenly falls to zero. In such a case, the unconstrained free turbine overspeeds and will be destroyed by centrifugal forces.
In other words, you do need a load of varying degrees, you cannot test an unloaded fully assembled free turbine engine.
As for efficiency, efficiency is a measure of fuel flow per unit thrust/torque, so a load is needed to get that measurement.
Based on the comments, I think you are asking: does the fuel burn get affected by loading on the gas generator shaft? And by loading, I think you mean the load through the accessory gearbox to drive things like oil pump, fuel pump, alternator, etc. Yes, those absolutely affect the fuel burn. Think about the consequences if they didn't. If you could pull 10x as much power out through the oil pump without affecting fuel burn, then you've created energy for free and violated the laws of thermodynamics. So yes, that affects the fuel burn. The question is, do it affect it much? Not much, but probably enough that you'd need to account for it. The amount of power drawn by the accessory gearbox is relatively small compared to the amount drawn by the power turbine. I would guess it's a few percent. But an error of a few percent in fuel burn calculations is probably not acceptable. So you'll want to account for it.