Slighty right at the start, and more severely when the suspension starts to rot. These initial changes should be very slight though, and are typically much less than tolerances between individual speaker elements anyway. Also, these changes should really not have any significant impact on the sound.
Well, what can happen is the material thats used for the "spring" can be a little stiff just after it comes off the manufacturing line, giving a slight taughter, constricted sound. Burning in is loosen this sufficiently.
One way of burning in high end speaker is to have the left and right channels facing each other and then reverse the polarity of one of them so that they are in phase with each other, you can then gradually ratchet up the volume.