Actually, as negligible as it may be, I believe that when the aircraft is pressurized at altitude, the net effect is that the air inside the aircraft will weigh more.
Reason being that on the ground, no pressurization, the weight of the air inside the aircraft is balanced by the buoyancy effect of displacing an equal volume of air at the same pressure.
At altitude, if the aircraft is not pressurized, then the weight of the aircraft is again balanced by the buoyancy effect (lighter air both inside and outside the aircraft). If the aircraft is pressurized, then the weight of the air inside the aircraft will increase and there will be a net increase in weight.
Of course this is all academic as compared to the weight change due to fuel consumption.