After coming across this article here, and reading the paper here I came away with the impression that the authors were saying that polar wander is being affected by climate change due to mass rebalancing as ice caps melt. As I understand it, the magnetic field of Earth derives from the liquid outer core, and due to the sheer mass difference between the core and any surface mass losses, combined with the lost mass being rebalanced isostatically, such a theory seems on first glance improbable. Could someone with more knowledge perhaps explain how the proposed theory operates?
The term ‘polar wander’ is ambiguous. In this case it's got nothing to do with the magnetic poles, but rather with the Earth's spin axis. If you think about a desk globe mounted on an axle, the poles under consideration here are the points where the axle intersects the globe's surface.
The paper is essentially saying that climate changes are changing the distribution of ice and water on the surface of the planet, and this in turn is shifting the spin axis slightly. You're right that the mass of material on the Earth's surface is dwarfed by the mass under the surface; however, if you look at the paper you'll see that they're talking about miniscule shifts in the axis. The changes shown in Figure 1 are on the order of tens of mas (milliseconds of arc) over a few decades. One mas is one 3.6-millionth of a degree, or about 3.09 cm as measured on the Earth's surface. So, at least at first glance, the changes don't seem too implausible.
You mention isostatic response as another concern, but this is a slow process which doesn't really come into play on the decadal time scales considered in the paper. Fennoscandia and northern North America are still rebounding isostatically from the last ice age.
Further to Pont's comments, there is often confusion about polar drift, which is two entirely different processes. The magnetic north pole is created by the rotation of the Earth's core, and possible differential rotation between core and mantle. This effect has historically shifted the magnetic north pole by about 50 km.
The geographic north pole, which is the Earth's spin axis, has seen significant shift since the end of the last ice age, mainly due to melting of the North American ice cap. As N. America got 'lighter' the geographic north pole moved towards Hudson Bay. Now, with climate change Greenland is starting to melt so that the polar shift has changed direction - towards the UK in fact, but the total measured effect over the last few decades is only a few tens of metres.