Leaving the burner lit when closing the valve on the tank or the solenoid will allow the stove to burn off most, if not all, of the propane that is in the line.
Dan, I have heard that claim, but I think it's erroneous for the reasons I mentioned above.
I would argue that if your line evacuates when you close the valve off and leave the appliance running, you probably should run a pressure check of your propane system. Because something has to replace the propane that is passing through the hose to the appliance, or else a vacuum would form. If it's not more propane from the tank, it would have to be atmosphere from outside the system being sucked in.
I do turn off the solenoid before extinguishing the appliance, but only to confirm that the valve closed properly. On our boat, the appliances extinguish almost instantaneously. And I've never noticed any lag when the valve is turned back on within a short period of time, suggesting the line remains full of propane.
Compare that to the lag noticed if the appliance hasn't been used for a while (days/weeks). But that is due to the permeability of the propane lines themselves -- a certain amount of the gas passes through the hose walls over time.
Either way, closing the valve is a good idea, and as Faster said -- because of human nature -- making it convenient to do so via a remote solenoid is the best approach.