Having owned a great number of headphones of which I have listened to out of the box with my test tracks and then put on "burn-in" for a day or two and then listened to again I can from my experience say that the effect of burn in exists to some degree but is not at all present with all headphones. Big changes are rare.
Audio Technica CKS77 (a very bass-heavy in-ear) is perhaps the clearest and most dramatic example.
Out of the box the bass was noticeably uncontrolled and muddy and the overall sound lacked clarity, detail and refinement. I almost though the pre burnt in CKS77 was defective or fake.
Come two days later the bass had tightened up a great deal (still on the boomy side though being incredibly bass-heavy) and the mids and highs had come forward showing more clarity and detail. The biggest change was the tightness of the bass. It would have been fun to compare a pre burnt in CKS77 to a post burnt in CKS77 side by side to really get a true representation of the overall change because the perceived change was so drastic.
From reading other peoples comments it's seem to be a rather consistent aspect of the CKS77 and the CKS line.
The thing is that we are influenced and conditioned by past experience and expectations to a bigger degree then we realize.
For example If you listen to a bright and bass light headphone for a long time and then put on a pair of more bassy and smooth headphones of roughly the same technical ability headphones you will feel the new headphones sound muddy and dull, but if you listen to that same headphone for a long enough time and put on the first headphone you will think it sounds anaemic and harsh. Our brain have a remarkable ability of adapting to new circumstances and make it the new baseline of which you form your opinions.
Most of the phenomena called burn in will be people getting used to the new signature to a large degree.
Unless the change is dramatic I doubt peoples ability to really differentiate between real sonic change and their brain changing. I don't really trust my own either.