Is "the earth moves around the sun" a case of Occam's razor? This question is a bit more meta and non-straight forward than it might appear on first glance.
I recently saw Why The Speed Of Light Is Unmeasurable, which is actually about why the speed of light cannot be measured one-way, but only for the round trip, and about the respective convention to say that the round-trip time devided by two is the one-way time.
Now, if we do not have a reference point one could think that we are not able to say whether the earth moves around the sun or the sun around the earth.
https://physics.stackexchange.com/a/10936/131815 explains if we assume that the earth moves around the sun, we can work with Newton's laws (which is nice). If we assume that the sun moves around the earth, we cannot use Newton's laws. Now, we could imagine that another - much more complicated - set of laws would be able to describe the phenomenons that Newton's laws could not explain under the assumption of "sun moves around the earth".
Is it a case of using Occam's razor to go with Newton and "earth moves around the sun"?
 A: 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.
A: 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 one of the above 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.
A: 
Is it a case of using Occam's razor to go with Newton and "earth moves around the sun"?

Yes, however there is a big caveat.
Occam's razor says that if two theories fit the data equally then the simpler theory is preferable. In this case there are two primary theories of gravity: Newtonian gravity and General Relativity. The two theories do not fit the data equally well in general, so in general you cannot use Occam's razor to choose Newtonian gravity.
You can only do so in the specific cases where the differences between the two are too small to matter. Luckily, this includes many scenarios, such as launching satellites into orbit or sending rockets to the moon or other planets. It does not include other cases such as predicting the precession of Mercury far into the future, the rate of clocks at different altitudes and speeds, the time delay for radar pulses sent to distant planets, gravitational lensing, etc.
This is important to your question because while Newtonian gravity is built around inertial frames where the earth moves around the sun (or rather the barycenter), it turns out that GR is perfectly content with coordinate systems where the sun moves around the earth.
