There are changes which happen to the Cu and dielectrics in the cables with time. The question is whether or not anyone can hear anything audible in the slight changes that occur.
Here are the four physical phenomenon I know about:
1) The Cu crystal grains that make up the wire will slowly grow. It's called grain coarsening. The larger Cu grains will have a slightly lower resistance than the smaller grains since there will be fewer grain boundaries to scatter the electrons carrying the current. If the Cu was oxygen free and was annealed after being drawn into wire, this effect will be extremely minimal.
2) High current such as that which might be present in a power amp driving huge bass woofers can cause electromigration, or movement of the Cu atoms at the grain boundaries which can slowly separate the grain boundaries. This leads to higher resistance. Electromigration occurs on the order of 1e5 A/cm2 in Cu. This is not going to happen in the wire of a headphone. Also, electromigration is almost unmeasurable in wires which carry AC current.
3) As with any dielectric, the transmission characteristics of the wire will change a bit as it is stimulated by a signal. However, audible frequencies are so low that transmission line properties hardly matter.
4) Kinking a wire will introduce all sorts of grain boundary separations and dielectric distortions, changing both the resistance and transmission characteristics of the wire. Ask yourself if you've ever heard a difference in a cable after accidently crimping it (assuming you didn't sever it).