I'm in camp amp needs to be on to pass full signal through output section of dac.
As for burn in, I absolutely hear it, this over many decades and all sorts of equipment. But in any case, it all becomes a non issue once components have a certain amount of hours on them. Problem arises when one doesn't believe in burn in and dismisses/banishes newly purchased equipment, I've had personal experience in buying equipment from churners who didn't believe in burn in, I don't let on a thing, they assume equipment lousy, may be willing to let it go at a lower price.
I don't even bother dealing with burn in much anymore, have a desktop or bedroom system I can use for burn in purposes. Speaker, speaker mods and amp output tubes are the only instance in which I have to put up with burn in.