I would think the ELR and the Volt would be similar. It appears as if the Volt displays a Propulsion Power Reduced (PPR) message to let the driver know only one of the two primary sources of electric power is available to supply power to the electric motor. Most often this is seen when the ICE is running in Normal mode and the car performs a maneuver requiring more power than the generator can supply (such as driving up long, steep hills). The depleted battery is unavailable, so PPR is displayed, and the car performs as well as it can using only generator-supplied power.
PPR may also occur following an FMM (Fuel Maintenance Mode). If you empty the gas tank while performing the FMM, the battery becomes the sole available source of electric power (there is no gas to run the generator), and the PPR message will appear on the display. Perhaps this is why engineers required a certain minimum amount of gas to be pumped into an empty tank to trigger an end to the FMM (wanted to be sure the car was carrying enough gas to power a battery-depleted Volt to the next charging or gas station).
The quoted message by Saghost seems to suggest that a PPR following an FMM might also limit the battery output and acceleration rate capability, thus maximizing the remaining battery range by reducing the rate at which power can be drawn from the battery. However, if you are not driving on a battery nearing the minimum state of charge, or under conditions that require more rapid use of the available power remaining in your battery - for example, driving up a long, steep hill, or accelerating rapidly - PPR should be little more than an irritating message on the display as long as you can use the remaining power to reach a gas or recharging station.
I suppose someone could drive a Volt with no gas in the tank long enough to trigger an EMM. During an EMM, the car is powered by the output of the generator. The generator can’t start if there is no gas in the tank, and thus the car most likely won’t operate. The same, I suspect, can be said of the ERDTT in cold weather. If the cold temperature requires running the generator, and if the absence of gas prevents the generator from running, the car most likely won’t run.
One could suspect that the same would apply to the ELR. It might well continue to run on battery only power, with no gas in the tank, until the first EMM - but the implication is that the Propulsion Power Reduced message would continue to be displayed, and most drivers would think it prudent to correct the condition that produced the warning message.