Relativity matters for GPS because the GPS clocks need to keep time so precisely. A nanosecond of error in the clocks translates to .3 meter of error in the location of a GPS ground unit.

Relativistic effects might matter slightly when it comes to making the clocks attached to the cable as accurate as those of the GPS satellites. The effects would be greatest for the clock at the upper end of the cable, where the counterweight is located. Depending on the mass of the counterweight, it might be 100,000 to 200,000 km from the center of Earth and going around once per sidereal day. That's about 26,000 to 52,000 km/hr. The relativistic gamma of 52,000 km/hr is about 1.000000001. In special relativity, that would be about .1 millisecond per day. But since the motion is circular, you need general relativity, and I'm pretty sure the cumulative effect would be a lot less than .1 millisecond per day.

EDIT:

Here's a good site for space-elevator design. I don't know which is more important, relative velocity or gravity.