A Spanish hifi site, Hifi Matrix e-mailed various speaker makers and the results varied from no running in to hours to we know the speakers do run in because audiophiles tell us so.
Various tests of woofers have found that the frequency response does change with use over a number of hours. But, if rested they will return to their original state. So burning in is only temporary and if speakers are used regularly remains, otherwise not. I have not found any such testing with tweeters.
Another forum which wanted to try two same brand headphones to see if there was burn in found that no two headphones new measured exactly the same. Then, it was debatable whether those differences were audible or not before burn in and once one was burned in and the other not, how could that result be used to show burn in with all headphones?
As for solid state hifi equipment, there is no evidence to support burn in causing audible differences, but the ear tuning itself to the new sound is as least as likely a cause of such burn in.