I am not quite sure what exactly you are wanting, here.
On Mach 3 you say you do not have the Z axis set up. However, If you are worried about it moving, it implies that it is fitted and working ??
You can try two ways - On config/ports and pins - turn the z axis off. I haven't tried this - the Z axis will not work, but whether Mach says - "You are trying to move the Z axis and have it turned off so I will not play" I don't know.
The other way is disconnect the step and direction wires to your stepper driver or whatever, and turn the power off to the drive. The Z axis will definitely not work, and since you are not affecting the "computer" side of the set-up, Mach will be unaware that any thing is wrong.
You MUST make sure, however, that the X,Y movements you make do not depend on a particular height of Z, e.g. to miss a clamp of anything - make sure z is fully up and out of the way.