Hmm. Interesting one.
Only things that come to mind are that the voltage regulator inside the gauge has failed in some way, although I would imagine this is combined with the rest of the dash electronics, or that the signal from the float in the tank to the dash is getting upset by additional loads on the electrical system when running, perhaps due to a dodgy negative connection to the vehicle body either somewhere near the fuel tank or under the dash area.
It might be that the float sensor shares a negative connection with the fuel pump, for example, as these are both located in the tank. A bit of voltage drop here could well upset the gauge without seriously affecting the fuel pump operation.
I'd check the wiring around these areas and failing that get a Tech2 on it. It's possible that a Tech 2 can look at the sensor readings seen by the dash and can see if that is also changing when the engine is started.
Just another thought. Try measuring the battery voltage when engine is both stopped and running at a fast idle. It could be that you have a faulty alternator that's generating too high a voltage and that's causing the gauge to malfunction when running.
Kevin