If the axis of the Earth was through Greenland, what would the north celestial pole be? If the axis of the Earth was through Greenland, what would the north celestial pole be?
If you assume that the shift was due to natural process, it would still point more-or-less towards Polaris.  (If the shift is due to magic or advanced technology, all bets are off.)

I think your question may be based on a false premise, that the celestial sphere is fixed relative to the Earth.  It isn't at all.  Over the timescales we're dealing with it's basically just fixed and unchanging.

The rotating Earth acts like a gyroscope and its axis points in some direction.  To a good approximation, that direction is fixed in space due to conservation of angular momentum.  So in the absence of external torques, the Earth just sits there and spins like a top.  

(To a better approximation, there are some external torques and the gravitational effects of the Sun and Moon causes the Earth's axis to slowly precess. It wobbles (a bit like a top slowing down) once every 26,000 years, actually chaning where it points to in space.  I think this is probably an irrelevant detail though, for your question.)

So, if the Earth's axis passed through Greenland, what would happen?  (Other than the climate changes due to the polar and equatorial regions moving around, of course.)

We have in fact seen something like this in the Earth's past, and it's called Polar Wandering. We observe that the Earth's north pole has moved relative to the Earth's surface and actually has moved quite a lot. But what happened was that the Earth's axis stayed fixed relative to the stars and the Earth's bulk shifted relative to its axis of rotation.

The True Polar Wander page referred to above has an excellent diagram: 

So, the answer is that if the shift occurred because of changes to the Earth's moment of inertia, the axis of rotation would still point towards Polaris.  If it happened due to outside influences (a giant impact, aliens, magic) it might wind up pointing most anywhere.