There is certainly real value in burning in electronics in order to weed out infant mortality of components, thereby improving the reliability of the product as it reaches the customer. Integrated circuits, whether digital, or analog, or mixed-signal (like DACs) exhibit a bath-tub shaped reliability profile, with a relatively higher percentage drop-out of parts during the early, infant mortality period, a low steady-state failure rate during the remaining life of the product, and then some increased failure rate at end-of-life. Different parts and technologies will have widely differing failure rates, but the bathtub-shaped curve characteristic tends to hold true. So, if an amp manufacturer runs the amp for some period of time before shipping, it will result in fewer infant mortality failures after shipping. You can also accelerate the time required for weeding out the early failures by running at elevated temperature. In fact, since it is impractical to actually run electronic parts for 10 years to verify that they are good for that long, or to verify their long-term, steady-state failure rate, the basic technique used for reliability testing is to run them at significantly elevated temperatures, as well as under the combination of high temperature and high humidity (such as 75-85 degrees Celsius, and 85-90% relative humidity. There are equations to translate hundreds or thousands of hours at elevated temperature and humidity, into years of operation at normal operating conditions.
In terms of whether the sound improves with burn-in, I am skeptical as to whether there is any difference for electronics, especially digital circuitry. But, I do buy the possible logic behind breaking in speakers, since it at least seems somewhat believable that the mechanical characteristics of speaker pistons, cones, and surrounds would change a bit with use. But why this always seems to be for the better (in reading high-end audio magazines) escapes me. It would seem to me that it would be just as likely for a speaker to get worse with use, as it is for it to get better. So, for every speaker review that talks about the speaker "really opening up" after 100 hours of use, you'd think that there would be at least a few that "really got sloppy" after 100 hours. Or perhaps it is the breaking-in of the listener that is actually occurring, as the listener gets familiar with the sound of the speaker, and starts hearing subleties that they didn't notice at first. (In other words, I am not doubting that the speaker starts to sound better after X hours of use - I'm just not sure if it is the speaker itself changing, or if it is the listener's perception that is changing.) I also can't recall ever seeing plots of speaker response done before and after 100 hours of break-in, that showed any measurable difference.
