Here we go again. Just when you think you have explained 'burn in' to death, along comes another 'it got better over time thread' with absolutely no basis except 'it sounded different to me'.
So lets start with 'burn in'. This is where electronic components (or more often complete systems) are run at elevated temperature to encourage early failures. Electronics suffer from infant mortality - either it breaks early on in its life, or it lasts a long time. 'Burn in' weeds out the components that were going to fail early. Important where warranty claims are an issue, in mission critical systems, and where replacement would be difficult. Absolutely nothing to do with 'golden eared' hifi enthusiasts.
About the only components in hifi that change over time are electrolytic caps, and not for the better. They have an electrolytic gloop inside that can leak or corrode as time passes. A warning sign is the cap case starts to bulge. I have worked on old valve equipment where the old electrolytics have exploded.
Mechanical systems can and do 'run in', so although I am sceptical about the magnitude, I can at least see a system where a speaker could change through usage. Amps and cables? no way. They are not Ford Escort engines.
The least reliable and most inconsistent part of the chain is between the ears of the listener. If anything is burning in and changing with time, it is the listener himself.