It is likely to be so low and so dependent on things that you can't easily quantify that you would have to determine it experimentally. For example, if the nut is cut such that it causes strain in the threaded shaft, i.e., making it bow back and forth, this could easily generate as much or more heat as the nut on the shaft.
Generally speaking, though, the heat generated will be expressed as: Energy = Torque' * Theta, where Theta is the angular distance turned and Torque' is the torque due solely to the friction between the nut and the shaft. However, I think that you're going to have an extremely difficult time separating out the torque due to the friction from the torque due to moving the load.
So... what is the application? Anything interesting?
Engineering is not the science behind building. It is the science behind not building.