I don't really believe in mechanical burn-in, I think its our ear that adapts. Last year I lost my hearing for several months, and I remember when I finally got it back every sound was excruciating. Talking sounded like shouting, I would jump back at the sound of paper rustling, playing violin was like taking a chainsaw to the side of the head. But after three days it sounded fine. I think that's an example of the 'burn-in' effect, although it might be a bit more extreme than that with headphones.
When people discuss their setups, they usually talk about things like the music source, their DAC, their amplifier, and their headphones. Some folks also discuss their cables, their power supplies, perhaps power filtering equipment. Some folks discuss issues with ground loops, noise, and other issues stemming from the connection of various components. Others discuss their recordings, whether they are lossy or loss-less, what the bitrate is, and so forth. I think that it's safe to say, that every individual piece of the components used to reproduce music, comes under intense scrutiny on this website.
This amazingly detailed analysis generally stops at the interface between the headphone, and the human listening to the music.
There is another chain of components involved that seem to be less frequently discussed, and even more poorly understood. I'm talking about our ears, the response of our ears, and our minds - which includes the understanding of what sound quality is, how to describe the various aspects of sound quality, our varied tastes in music, and so on. Of necessity, this also includes the age of the listener (which has an affect on how well we can hear things), as well as the "wear and tear" on each person's ears (those who've listened to loud music for years are less able to hear minor aspects of sound quality, vs those who have been kinder to their hearing).
It is my assertion that these details are what causes many disagreements and arguments over what we are hearing. We cannot ignore the fact, for example, that a man of my age (50+), who also has a provable hearing loss, simply is not able to hear the same detail as a man of say, 18 years old, who hasn't damaged his hearing. Similarly, a person much younger than I, who hasn't spent years of his/her life as a musician, may not have the same level of music appreciation that I do.
This is not a put-down. I'm just illustrating the fact that people's perceptions of music and sound quality differ. We often discuss such topics as sound quality without any regard to comparisons of the listeners. We like to pretend as if everyone has the same level of music appreciation, the same level of hearing ability, and the same ability to identify and compare very slight differences and aspects in sound quality.
This simply is not the case. People are different. I think it would be more accurate to say that everyone has their own individual perception of what sound quality means to them.
My observations on this thread (and others), years of being a musician, and years of listening to music, demonstrate that my perception of sound quality changes, from one day to the next, and from one set of equipment to another. It is horribly difficult to establish an accurate basis of comparison, just for one individual - nevermind attempting to establish a basis of comparison between any two human beings on this planet.
That being said, we still attempt to compare things. We try to identify and quantify such things as "burn in" - whether this is real or imaginary. I have previously stated my beliefs on this thread. My observations are relevant to individual pieces of audio equipment, and not to the human beings using them. Most of us have discussed these issues on this thread, without paying any attention to the human factors involved. And therein lies the problem: we're comparing the wrong things.
Outside of the fact that tubes actually do "burn in", headphones may "break in" due to mechanical wear, and solid state components don't change much (if at all) with use, we still need to discuss the real variables in this equation: our ears, our minds, and our perceptions. In my opinion, those are the things that vary far more than any of our audio equipment. And since it is very difficult to establish a basis of comparison, we're arguing over something that we'll never be able to prove.