Does GPS correct for continental drift, and if so, how? (I'm mainly interested from the perspective of consumer devices such as smartphones.)
No, GPS does not 'correct' for continental drift per se. GPS can be (and is) used to measure drift. Drift is accounted for in the model of the earth used, aka datum or reference ellipsoid. GPS uses the World Geodetic System, or WGS, and most units report coordinates in the initial version established in 1984 (aka WGS84 coordinates). That model, and others like it, can be and are revised from time to time. Revisions can account for drift as well as more accurate modeling and measurements.
Technically, it's not the GPS coordinates that change, it's what's located at those coordinates that does since the ground is what's moving. WGS uses the theoretical center of the earth's mass as its origin, so everything on the surface 'floats' relative to that. Some models/datums are tied to a specific plate, in which case everything else on that plate should keep the same relative position (give or take - there's always local fault lines), but things on other plates would 'move'. Here's one of many discussions you can find on the topic.
Now since you're talking about consumer devices and in particular smartphones, the scale of drift occurrence is far beyond the accuracy of such a device. Drift is like 2-4 centimeters per year (disclaimer - figures differ, plates are moving at different speeds, etc.). There are other questions here already on GPS unit accuracy that delve into applying corrections and such, but we'll go with a nice round number of at best 3 meters. In other words, with a consumer device the maximum error for drift within a hundred years is about the same as the potential margin of error between any two given readings from the device.
Not unless you use special software; it only makes sense if your device is firmly attached to earth for year-long periods; see sec. 10.1.1 ("Site Displacement Modeling: Tectonic Plate Motion") in the BERNESE Manual:
Station coordinates are changing in time due to the steady movement of tectonic plates. Figure 10.8 shows the present day major tectonic plates. This movement must be taken into account in GNSS analyses. Station coordinates (especially of reference sites) should therefore always be propagated from the reference epoch to the observation epoch based on corresponding station velocities. This ensures consistency with the IGS satellite orbits and prevents network deformations induced by moving plates.