Some equipment clearly "burns in". Tubes are the best example. They are metal glowing red hot so it makes sense they need time to reach equilibrium. Electronics are very small circuit paths in semiconductors and so I can see them rearranging some atoms along the way. But the time needed to burn it is greatly exaggerated.
It is very convenient for a manufacturer when a customer calls up and says "I don't like the sound, I want to return" and the manufacturere can respond "How long have you listened. This ___________needs 300-400 hours to burn in." So the customer hangs up, keeps listening, gets used to it, and figures out 400 hours will take 6 months and after 6 months it is much less likely he will return...oops, lost the box if nothing else.
And I'm skeptical about many things burning in--copper wires for one. speaker wires, interconnects, and power cords. The atomic structure of copper doesn't change with low level signals running through it. Some of these same manufacturers try to tell us they have found directional copper.
OTOH, sometimes "burn in" can include the consumption of some alcoholic beverages and on that 10th night of listening with just the right blood alcohol level, this _____________ is now starting to really burn in and sound great.
Jerry