I have a 696 diagnostic fault code on my EF Falcon XR6. That translates to:

Trans Solenoid S6 Fault - EEC Cannot control the solenoid. Can be caused by open circuit, or short circuit to Ground, +12V

The information I can find on the S6 solenoid is:

S6 is a normally open On/off solenoid that sets the high/low level of line pressure, Solenoid off gives high pressure.
The primary regulator valve (PRV) (refer to figure 3.19) regulates the transmission line pressure (or pump output pressure). This valve gives either high or low line pressure depending on whether S6 is switched Off or On, When S6 is switched On, S6 pressure is applied to the PRV moving it against spring pressure and opening the line pressure circuit to the pump suction port resulting in reduced line pressure. Low line pressure is used during light throttle applications and cruising. Heavy throttle will cause S6 to switch Off and thereby cause high line pressure.
This stepped line pressure control has no detrimental effect on shift feel because all shifting pressures are controlled by separate band and clutch regulator valves, and the output of S5.

Based on this info, if S6 is not controlled by the ECU, line pressure will be permanently set to high. I haven't noticed any significant transmission issues. It will occasionally thump on a light throttle shift. But the difference in line pressure between Econ and Power does not appear to have been affected. So, I am happy to run the car with this fault until I have the time to get it fixed.

But, am I doing the car any damage by running with the line pressure permanently set to high? For example, would that generate extra heat in the transmission? Thanks for the help.

I'd get the solenoid checked ASAP. You run the risk of seals leaking/failing and parts in the box not being cooled properly with prolonged use.