I think the whole idea of "burn in" came from the tube and speaker world. I never understood it with tubes, since the elements are in a vacuum, and I never noticed any difference in the sound of a tube amplifier after using it for a hundred hours. But what do I know. They certainly can burn out and stop working, so obviously they change somehow. And different tubes sound different in an amplifier, but it's because they're made differently, and they're amplifying the actual sound that's being produced by your guitar or turntable or whatever.
Speaker cones used to all be made of paper and rubber, and paper and rubber change in all kinds of ways for all kinds of different reasons, so you can definitely hear the difference in a speaker after it's been used for a while. But then the latest stereo speakers I bought have "woven aramid-fiber" cones (some kind of carbon fiber composite). So do those "burn in"? I have to think not, being carbon fiber and all, but again, what do I know.