Does new chain on old cassette damage chain?

The main problem with an old chain is that it's stretched. As a result, it won't sit snugly on all the covered teeth of the sprocket, but only on the ones with the main load, and as you pedal the contact point slips a bit. Just fractions of a millimeter at each tooth, but with the considerable load and many thousands of revolutions, that's what wears out both the chain and sprocket much quicker than a properly fitting chain does (which doesn't slip on the sprocket but simply “peels off”, the only friction being inside the chain-link bearings themselves).

An old sprocket can cause a similar problem too, but only if it's quite extremely worn. Usually, the more pressing problem will be that the chain doesn't properly shift between gears anymore, or shifts/skips when you don't want it, because the tops of the teeth and the fine-engineered ramps on the sprockets' sides are compromised. This skipping and mis-shifting under load does then again put wear on the chain, but mainly it's just super annoying for the rider. It's not like the stretched-chain issue, which stealthily grinds on the parts while the gears still seem to work fine.

So, no, using an old cassette with a new chain is generally fine, as long as the gears are working well.