Speakers are mechanical devices and so their properties change slightly with use. The surround and suspension components and their characteristics will subtly alter with time and use. Same for cartridges.
As to electronic parts, and I am speaking as an electronics engineer... they do not require burn in or conditioning or anything else. An exception to this is the "forming" of electroylitic capacitors that have been out of service for many years but that is another issue.
To all those that believe they do, then I say ask yourself why the "change" in sound that you claim to to hear is always for the better. Why would any change (if it existed) not be for the worse ?
Ultimately if you are a "believer" then yes, the sound will improve for burn in.
Absolutely right Mooly.
When I took delivery of my Nait 5i, it sounded the same from the time i took it out of the box to now. No difference at all. However, I also have a Roksan Caspian amp. When I fire that up, it 'appears' to take some 20 minutes to come up to full bandwidth. If I had nothing better to do and if I was sad enuff, I could sit there and listen to it open up ( I have done it once ).
As regards speakers, again Mooly is right. Taken fresh from the box, they don't sound right. The 'burn in' period is to allow the adhesives that secure the front surround and the inner spider to flex a little. When speakers are designed, they are done with well worn drive units so that the parameters are constant.
Electronics burn in?. Not convinced it needs it. Loudspeakers?. Absolutely true.