Does an unloaded amplifier use more power at a higher volume?
It is using power and producing heat, but probably not as much as if it was driving the outputs to full power. The amount of power will depend on the type of amplifier. A Class D amplifier may use very little power when quiescent, a Class A amplifier will use about as much power idling as when it is producing full volume output. Most consumer electronics will use Class D or Class AB amplifier for the output stage, so it won't use much power compared to when it's driving speakers to full volume. Class A for a power amplifier is mostly the province of audiophiles and audiophools. 

I don't think it typically makes much difference where you leave the volume control- that's an attenuator that's back of where the heavy lifting gets done. I guess if you're plugging and unplugging input cables you might damage your speakers from the loud hum and buzz that occurs if the volume is cranked up to 11.