I guess it must Alex, though given the weights involved in our model control surfaces it is probably of minimal effect compared to the aerodynamic forces.
The typical model control surface is front hinged and not mass balanced about the hinge line, so the weight tries to make it fall. The force will be a multiple of the G load. The servo has to counter that force, so it does have hold the surface weight against G load. In some cases this will assist the servo, in others it will add to the power required.
Consider a flap that is down, the airflow is trying to push it back up to neutral and its weight is trying to pull the flap down, the servo has to provide enough force against the balance of the flaps aerodynamic less its weight forces. If G is increased, the weight pulling the flap down will increase, thus helping the servo work against the aerodynamic force that is trying to push the flap back up. But you can easily understand that the aerodynamic force is much stronger than the light weight of the flap, so the aerodynamic force still dominates the equation.
On the other hand, consider up elevator, the servo has to hold up the weight and hold it up against the aerodynamic force trying to push it down, again an increase in G will increase the weight but this time it adds to rather than subtracts from the aerodynamic force, but once again in most cases the weight even under several G is likely to be a fraction of the aerodynamic load.

Something that occurred to me with all moving tails is not so much the aerodynamic load which can be minimised, but the rotational momentum due to the weight. My F-86 tails are not balanced about the pivot, are light, but the servo is always buzzing slightly against the weight it has to hold. On my F-100 the tails are mass balanced about the pivot, that took a fair bit of lead, so when you move the tail it takes a good bit of force to get it moving, and then to stop it at the correct position. I haven't yet done Oli's calculations for the tailplane, but if it is well designed and made it should present a surprising small aerodynamic load but that doesn't take account of the weight which in the case of a mass balanced AMT could present quite a load for the servo trying to start and stop it, and may become the dominating factor or at least significant enough to warrant being calculated in addition to the purely aerodynamic force.

H