Burn-in for speaker drivers is well-documented. Changes in Thiel-Small parameters can easily be measured. Interesting thing is, while there are changes, they are somewhat proportional. I always burn-in drivers with 24 hours of low-frequency signal and then stick them back in my measuring jig and recalculate TS parameters. After this, there really isn't any measurable difference, and the TS parameters don't change significantly, or at least they change below my measuring capabilities.
As for amplifiers and what not, I always "burn-in" new gear, but for me this is more a matter of adjusting operating points. I try and simulate the final operating environment of the amplifier as much as possible (i.e. cased up and on my rack). I feel this is worthwhile, but YMMV.
For turntable cartridges, the suspension definitely settles in over time. This is pretty obvious, as you have to fight with tracking weight, and other parameters during this time. Again, pretty measurable changes. But then again, these parameters also change with temperature and humidity. I have to readjust my cartridge every 3 months or so, I've found.
Other than that, who knows. I hear things but usually I'm changing more than one thing so it's hard to pin down. I tend to say, why not do it...