Does the geographic North Pole move? The magnetic North (or South) Pole moves due to activities in the inner of Earth. And earthquakes can tilt the Earth's orientation a tiny bit. But can and does the axis' orientation relative to the surface change too? A major impact could surely do it, but could the Earth's inner activities?

I don't mean precession, nor tectonics or continental drift, but the movement of the rotation axis. Could for example the geographic North Pole move to Greenland? (Not Greenland moving to the NP)
Yes, it's called polar motion.

The rotational pole moves continuously, as you can see from the right-hand side of this figure (below) by the Earth Orientation Centre (EOC) and the International Earth Rotation and Reference Systems Service (IERS). The figure shows about 650 days of time; mjd is modified Julian day and time goes along the locus in the polar motion diagram.



The left-hand side of the figure shows Length of Day, and I think that's chiefly why they are making these refined polar measurements — to keep track of leap seconds, etc.

It's been established, by Fong et al. (19996), among others, that earthquakes could change the earth's rotational axis, by redistributing mass on a large scale. Their table 1 shows [the sort of effect that individual earthquakes theoretically have — edited after comment from David Hammen] on the length of day; $\Psi$ is the excitation vector:



The EOC's website lists other geophysical excitations, among them:


Atmospheric angular momentum
Oceanic angular momentum
Hydrological excitation function
Axial angular momentum of the core


References

Chao, Benjamin Fong, Richard S. Gross, Yan-Ben Han (1996). Seismic excitation of the polar motion, 1977–1993. Pure and Applied Geophysics, September 1996, Volume 146, Issue 3, pp 407-419. DOI 10.1007/BF00874727

Update: This article on some new research in this field is worth a read.