"Does the device (heater/grill) burn more of the gas if the tank's valve is open all the way?"
No. The tank valve is to allow you to disengage the tank from the feed line and to provide a safety cutoff valve in case the device's valve assembly fails to close when it's not using the fuel. Once it's firmly attached, you should open the tank valve all the way. The device itself will have a regulator and its own operating valve, and that regulation unit will be expecting full line pressure. If you use the tank valve to restrict the fuel supply, the device won't work correctly (and depending on the regulator on the device, it may not work at all). It'll use less fuel, but it won't do what it's supposed to do. To use your example of a gas grill, closing off the tank valve will cut down on propane usage, but it'll also prevent the grill from heating to its full capacity. For your heater, it won't generate enough heat to keep the room at the thermostat setting.