How detectable is the error in special relativity experiments because of general relativistic effects at the surface of the Earth? I believe that experiments done on special relativity in a laboratory at the surface of the Earth usually do not consider the effects of general relativity (since the gravitational field at Earth's surface is weak). Of course, errors shall emerge from the approximations, but is this error detectable at all? How much do the measurements deviate from the theoretical predictions because of general relativistic effects and does laboratory equipment have enough precision for these kinds of deviations to be detected?
 A: Nice question.
One way of stating the equivalence principle (e.p.) is that locally, spacetime is flat, so that special relativity is valid. Therefore any experiment in a small enough laboratory, if the apparatus is in free fall, is predicted by GR to give the same results as in SR. For example, SR predicts that if we release a brass ball and an iron ball at rest relative to one another, they will stay at rest relative to one another. This has been tested in Eotvos-style experiments, and the null results confirm SR and the e.p., but they are not specifically tests of general relativity.
If the experiment is local but the apparatus is not in free fall, but is anchored to the earth, then the e.p. predicts that the results are the same as in a rocket ship in outer space that is accelerating at g. An example of this type of experiment is the Pound-Rebka experiment. So this experiment as well is only really a test of SR+e.p., not specifically GR.
To get a real test of GR at the surface of the earth, the experiment needs to explore a region of spacetime at the surface of the earth that is large enough so that different parts of it do not all have the same acceleration. Tests that fall in this category include the Hafele-Keating experiment and the detection of gravitational waves by LIGO/Virgo.
