I'm going to jump in late as the map is very cool (I'll use it as a resource) but I am not sure it actually answers the question correctly (in a technical manner) despite all the upvotes.
I think the question and answers are confusing the maps, the gps, and even orbital oscillations of the planet in relative space or about an axis.
The Sendai earthquake a couple days ago has shifted the earth on its axis and moved the island 8 feet.
I'd like to split it into two. First the 8 ft and then then axis part.
- The island has only moved 8 ft relative to a static non-moving place (which does not really exist) on earth, somewhere. This makes almost not one iota of difference to the GPS system. The GPS will report a differing location for say a rock on the island, but the rock relatively moved and not the global positioning system measure. I am hedging a bit as technically the fabric of spacetime would have changed, which would have altered the GPS satellite paths relative to the earth, but it is absolutely immeasurably small (and then some) and massively insignificant and unmeasurable but if mass shifted around our geoid this must happen, but in reality it is a number so small I cannot even imagine it. Although the mass of the earth has stayed the same the mass is now organized slightly differently so it must have altered the fabric of spacetime, which will alter the GPS satellite orbits, but by such a miserably insignificant amount. We are talking a number that is so low I could not write it down.
This is the best I can find for (1) but you have to use you imagination. The earth is actually a more complex shape than this (a geoid) so it is not a perfect sphere and hence the fabric of spacetime is not warped so evenly. If you move mass around the earth then the geoid changes, and hence the fabric of spacetime changes, and hence the satellite "path" is altered.
Einstein eventually identified the property of spacetime which is responsible for gravity as its curvature. Space and time in Einstein's universe are no longer flat (as implicitly assumed by Newton) but can pushed and pulled, stretched and warped by matter. Gravity feels strongest where spacetime is most curved, and it vanishes where spacetime is flat. This is the core of Einstein's theory of general relativity, which is often summed up in words as follows: "matter tells spacetime how to curve, and curved spacetime tells matter how to move".https://einstein.stanford.edu/SPACETIME/spacetime2.html
I am using mass as opposed to matter (incorrectly) but I think you get the drift. It is the bold part that draws me to this conclusion.

- Now to deal with the axis part. If you chuck a rock to the other side of your yard I would assume this shifts the earth on its axis but who cares. Certainly not the GPS system. Ditto with the earthquake, as any significant axis shift that GPS could not account for, caused by an earthquake, would have to be so massive we would not be here to take the incorrect GPS measurements. So just as 1, yes technically it did but so does everything and it just does not add up to anything. Even when it is something we need to account for such as a massive earthquake that does not wipe us out the GPS system will correct for this small alteration.
The Earth's rotation axis does not point exactly along the axis of its angular momentum because it does have a tiny amount of lopsidedness to its distribution of mass.
https://van.physics.illinois.edu/qa/listing.php?id=1031