things that get hot, especially things that glow, will change with use, thus burn in. Materials do not change appreciably at room temperature or with the relatively tiny currents put through them by generating music.
If a manufacturer is designing components (other than tubes) in a power supply, amplifier circuit , or especially a low power component such as a DAC to get hot, then that component is probably not for me. I guess some amplifiers might sometimes be an exception here (high currents) but preferably not.
Copper is not directional. To make a wire directional you have to add electronic components. If a cable is shielded and grounded only on one end (a good way to avoid ground loops), it might be better to ground on the source end...but if both components are grounded to the same circuit, it really shouldn't matter.
Saying that a component needs "burn in" is a way of saying "listen to it a while before you return it" that sounds like it has a reason behind it.
Jerry