Headphone and speakers? Yes; those suspensions have to break in, and if properly engineered the parameters should be optimal after break in. I've had multiples of the same headphone in hand before (1 new; 1+ well used); some models show a more striking change than others. The HD650 and L3000 both start out somewhat dull and uninspiring (not bad, not missing any large chunk of frequencies; just mediocre) but develop their really nice and dynamic sound over time. That's a pretty striking improvement. On the other hand, will a headphone that starts out sounding bright and bass-anemic turn that around fully? Probably not...
With phono cartridges -- there's some, but it seems less in magnitude than with headphones/speakers. The cartridges I've heard right out of the box (all nice models) sounded at least close to excellent from the first drop, or very soon after (a few to several hours max). That's good since you certainly don't want to waste a cartridge's precious lifespan on burn-in.
With solid state electronics? I'm at least somewhat skeptical. These days if I hear a new component that I don't dig right of the bat, then I'm not going to wait around for a magical transformation. I once had a headphone amp repaired by a guy that wired the output caps wrong -- and then claimed that "it sounded good now, but it'll take some time for those caps to break in". It sounded horrific! After digging around I figured out that the way they were wired (WAY too small a cap in series on the outputs), the amp had a 6dB/octave low frequency rolloff starting at 1kHz! Certainly, burn-in gets blamed/credited for a lot of other forms of stupidity.
We may seem to get a little crazy with the burn-in on Head-fi, what with stuff like "the bass finally appeared after 700 hours". But that's nothing compared to what goes on in high-end 2ch, e.g. $13,000+ racks that need 300+ hours of burn in and -- according to the manufacturer -- supposedly will "sound awful at times during the process". Talk about BS 
Edited by mulveling - 2/11/13 at 7:12pm