If it does it in all modes EXCEPT when the dash vent is shut, then I don't see why it would have anything to do with the cabin temperature sensor fan, which runs constantly, and would seem to be physically unrelated to the state of the mode control motor.
I am having trouble trying to understand the nature of "clicking noise". Is it a slow click, like a clock ticking, or is it something fast, like a buzzing sound? Does it change rate directly with the blower motor speed?
I have two ideas:
1) The vent control door may be loose, and flapping around when it is open. If the vent door is loose, it should affect the way air comes out of the dash vents, possibly making the volume erratic. This noise would probably be something less than regular in its rate. You could do a simple test, by sticking a ruler in through the dash vent to try and hold the vent door steady when you hear the clicking noise, or just look at the vent door with a flashlight to see if it is steady. If it is, then you could remove the glovebox to view the smoothness of the mode cams and tightness of he various linkages as the mode is changed. Of course, you can watch for anything odd as the unit operates in a clicking-prone state. If you think the heater-evaporator needs to come out, indicating you think there is a problem inside the box, you may change your mind when you find out what is involved, even if you let a shop do it.
2) I don't really know if this is a problem that could affect an NSX, but I have seen it on several cars. I did remove an NSX evaporator, and there was trash in it, but I don't recall any in the blower. Leaves sometimes get inside blower housings, and the blower makes a noise like a playing card against the spokes of a bicycle wheel. However, I don't see how this could happen in all modes except when the vent doors are closed.
Not being able to look at your car in person, I am just trying to give you some ideas as where to start looking. Good luck.
Regards,
Bill
P.S.--I just went to look at the schematic for the Climate Control, and saw and remembered that there is a separate motor for the vent door. This motor might be a problem if you see erratic vent door operation, or there could be a problem in the control unit itself.