Is "the earth moves around the sun" a case of Occam's razor?

Yes. Geocentric models of the solar system such as the Ptolemaic system required very complicated and ad-hoc geometric mechanisms to explain the observed motions of the planets. Kepler's laws of planetary motion, which introduced elliptical orbits with the sun at one of the foci, were an enormous simplification. Newton's law of universal gravitation then showed that Kepler's three laws were a consequence of a single fundamental law, which also explained other phenomena, such as the motion of the moon, the orbits of Jupiter's moons, and the tides. In each case more observations were explained with fewer assumptions.

I don't think it is directly related to Occam's razor, this problem is more related to the Mach's principle, it roughly states that

The universe, as represented by the average motion of distant galaxies, does not appear to rotate relative to local inertial frames. Or

An isolated body in otherwise space has no inertia.

By assuming these we can say whether an object is rotating or undergoing circular motion(with respect to the average motion of distant galaxies). But Mach's principle is indirectly related to Occam's razor. It's not like we need to use Occam's razor for everything, if we use it for a general case we can deduce many simple things from that.

For example, GR(General Relativity) and Brans–Dicke theory both are in agreement with our experiments. Doing experiments where both give different results is extremely hard. But GR is simpler compared to Brans–Dicke theory, so we chose GR as the mainstream theory for Classical gravitation. The reason we chose GR is Occam's razor. But both GR and Brans–Dicke theory assume Mach's principle (in fact Brans–Dicke theory was developed such that it allows a stronger version of Mach's principle), so both theories say that we have to check the motion of an object with respect to the average motion of distant galaxies. There may be more complicated theories than these 2 which don't accept Mach's principle, but we chose the simplest which is GR. So after assuming GR we can't say that the sun revolves around the earth.

Occam's razor is not needed to explain some specific situations, it is needed to explain the fundamental laws of the universe.