break-in for amplifiers is definitely nonsense... you should not be "hearing" an amplifier, because it should be acting like a "piece of wire with gain". for speakers, it is usually nonsense too... there may be some changes in the suspension, especially with paper cone drivers, but it's not that much. one thing i have seen mentioned that's definitely true with paper cone drivers is that it's not a good idea to use single frequency sine waves for long periods (more than a few seconds at a time) because standing waves on the cone can set up rings of weak spots in the paper if the frequency is a multiple of the mechanical resonance of the cone itself (usually a good crossover can prevent this, because such a tone would be outside the normal frequency range of the driver). these days there are very few drivers that have paper surrounds. most low frequency drivers have some type of synthetic material in the surround, midrange drivers are usually either small cone drivers, or midrange horns using a synthetic diaphragm, and tweeter drivers use a synthetic or metallic diaphragm. the range of motion on mids and tweeters are much smaller than for woofers, so the surrounds on them don't have a tendency to change with use. since most woofers these days use synthetic material for surrounds, the time when the sound of a speaker is changed the most by physical changes in the surround is when the surround material starts to wear out. plastics and rubber materials begin to stiffen and get brittle as the compounds they are made of begin to break down.
so, if a speaker has a paper cone with a paper surround, yes, it may need some "break-in". but with modern drivers using synthetic materials for the surrounds? no, the changes in the surrounds won't be enough to notice, since the materials are very pliable to begin with.