My own commentary on the matter to date may be found here:
Geocentrism: Not at All an Infallible Dogma of the Catholic Church
The key to that whole commentary is the teaching of the Popes that the Holy Spirit did not put details about the physical universe in sacred Scripture:
There can never, indeed, be any real discrepancy between the theologian and the physicist, as long as each confines himself within his own lines, and both are careful, as St. Augustine warns us, "not to make rash assertions, or to assert what is not known as known."(51) If dissension should arise between them, here is the rule also laid down by St. Augustine, for the theologian: "Whatever they can really demonstrate to be true of physical nature, we must show to be capable of reconciliation with our Scriptures; and whatever they assert in their treatises which is contrary to these Scriptures of ours, that is to Catholic faith, we must either prove it as well as we can to be entirely false, or at all events we must, without the smallest hesitation, believe it to be so."(52) To understand how just is the rule here formulated we must remember, first, that the sacred writers, or to speak more accurately, the Holy Ghost "Who spoke by them, did not intend to teach men these things (that is to say, the essential nature of the things of the visible universe), things in no way profitable unto salvation."(53) Hence they did not seek to penetrate the secrets of nature, but rather described and dealt with things in more or less figurative language, or in terms which were commonly used at the time, and which in many instances are in daily use at this day, even by the most eminent men of science. Ordinary speech primarily and properly describes what comes under the senses; and somewhat in the same way the sacred writers-as the Angelic Doctor also reminds us - `went by what sensibly appeared,"(54) or put down what God, speaking to men, signified, in the way men could understand and were accustomed to (Providentissimus Deus 18; my emphasis).This teaching was reiterated by Pope Pius XII in Divino Afflante Spiritu 3. The attempts I have seen to bolster the claim that Leo XIII and Pius XII couldn't really be talking about geocentrism here are ridiculous. Indeed, the apparent motion of the sun with respect to the earth is the classic example of phenomenological language that is, "in daily use at this day, even by the most eminent men of science". Bottom line is that if the Holy Spirit did not put such details in sacred Scripture, it's not a matter of faith.
I am most concerned with the theological implications of holding geocentrism as a matter of faith. As I argue in the article cited at the top of this entry, I believe that it results in a de facto denial of the Catholic Church's indefectibility. But I have examined the proposed scientific evidence for the view and find it implausible, to say the least.
Here are some good sources looking at a variety of aspects of this question:
Geocentrism Disproved: How Newton's Laws Prove the Sun Orbits, by Ken Cole
As the Universe Turns: Is it physically possible for the whole universe to orbit the earth?, by Gary Hoge
Dialogue on the Center of Mass of the Universe: Why the earth can't be the center of mass of the universe, by Gary Hoge
Dialogue on the Center of Mass of the Universe, Part 2: Why the earth can't be the center of mass of the universe, by Gary Hoge
Geocentrism: Was Galileo Wrong? by Dr. Ethan Siegel
Flogging a Pink Unicorn: Why Modern Geocentrism is Intellectual Blancmange, by Dr. Alec MacAndrew
Here is some great commentary by a couple of science-savvy homeschooling moms:
The New Geocentrism
Up-to Date Cosmology
Relative Claims to Absolute Truth
Geocentrism -- where's the physics?
A final note on geocentrism
Also see Dr. Jeff Mirus', Galileo and the Magisterium: a Second Look and Dr. Thomas Lessl's The Galileo Legend and Dave Armstrong's "No One's Perfect": Scientific Errors of Galileo and 16th-17th Century Cosmologies Rescued From Inexplicable Obscurity.