Not really. The axis of the earth's rotation is not affected by plate tectonics and the star map is recording where the earth's northern axis is pointing as precession slowly moves it around a circle. The star map could certainly move around or get broken up by plate tectonics over that timescale but it's not really aligned with anything in a meaningful way so that doesn't matter.

The stars themselves will move - relative to us - and eventually some of them will disappear but nothing much is expected on that front for much longer than 10ky timeframes.