Well, there's a very simple test you could do. Next time you get a new pair of speakers, play one for a week or so, then use an A-B switch.
In a direct comparison, there's a small audible difference. Is it important to a consumer? Not really. The speaker will perform to it's parameters one way or another, but if you are designing a ported system pushing cone extrusion to the limits, it very important.
The question was, do speakers burn in? Yes they do. Is it audible? Only in the controlled situation.
Is it important? Not to a consumer. How do you burn them in? Play em.
This is my last shot in this ping pong.