There seems to be a mess of concepts and effects on this subject.
Some people seem to think that audio burn-in is like a car engine, where you're not supposed to over rev for the first 1000 km, for fear of reducing performance for the rest of the engine's life (something which never made sense to me...)
Having been among audiophile crowds since childhood, my experience is that all audio equipment has a burn-in (electronics) or break-in (mechanics) period. This is defined as simply the time it takes for the device to sound close to how it will sound for "the rest" of its life, as opposed to how it sounds straight out of the factory.
From my experience, with examples out of memory:
- electronics are rather quick, 10h tops (eg. Fiio E9, Yamaha RX-V1500);
- loudspeakers (dynamic*) take a bit more, maybe some 10-20h (Monitor Audio Silvers, Elac 200s);
- headphones actually seem to take the longest, some 30-50h (HD595, MX880, KSC75).
This is not to say that there aren't changes after this time, they'll just be more subtle and take longer to notice.
The signal used for burn-in does not seem to make any difference. We just hook them to a FM radio 24/7. We've tried noise, burn-in tracks from test CDs, etc. Don't seem to speed up the process in any way.
Interestingly, the changes seem to be similar. Bass and treble extension improve, dynamics improve, treble quantity decreases. Bass quantity increases on some devices, and decreases in others. Because of this, these changes are not always improvements -- for example, I prefered the KSC75 sound out of the box. The changes are also very slight, I have never heard sonic signature changes, for example, and I have a hard time believing some of the things people write about "miracle bloomings".
This burning-in effect is often mixed up with other effects, such as psychoauditory adjustment, which might explain alot of these "miracles" (I tend not to listen to equipment while in the break-in period, so in my case this is not a factor).
Another factor I have not seen discussed is the operating temperature. Solid state electronics tend to sound worse when they're cold; sound improves after some 30 minutes, no news there. But something similar happens to transducers as well (but in a much shorter timeframe).
*Elac's AMT tweeter took some time to get treble harshness under control, some 40-50h IIRC.