Q:

What happens if a geocentric model of the world were correct?

For a fantasy, I need to know how a world similar to Earth would exist in a geocentric model. 
1) I would assume the sun would have to be a lot smaller. I'm okay with artificial stars (hand-wave that stuff away with magic). But I would like to know if that makes an Earth-similar planet impossible. (I don't need other planets in this model.) For example, how would it effect... 

seasons and climate 
length of day, month, or year
sunrises and sunsets 
gravity
constellations and/or navigation 
any huge effect I don't have enough science to anticipate 

2) How would I manipulate my universe's model to make it more Earth similar if those things are completely off? It doesn't have to be exact, but I need a temperate climate with pretty normal seasons and climate zones. 

A:

The first thing we must understand is that from a purely kinematic point of view, the heliocentric and geocentric models are both equally correct within the accuracy limits of astronomical instruments available before the, say, 16th century. There was no way for an astronomer who lived before Tycho Brahe to bring serious arguments in favor of one or the other. (That's why Galileo Galilei had such trouble with the astronomical establishment of the day -- he simply did not have any good arguments to bring in favor of his pet theory.)
The problem is that from a dynamic point of view, for a geocentric model to be correct it is necessary to abolish the law of universal gravitation: and this means, of course, that the universe in which a geocentric model is correct from a physical point of view has vastly different physics from ours. Whether "a world similar to Earth would exist" in an universe with vastly different physics than ours is not something that anybody but you can answer.
How would it effect...

Seasons and climate:
We don't know. The law of universal gravity doesn't work in your world, so we have no idea how wind works, how the water cycle works, the lot. By the way, how does fire work in a world where the law of universal gravity does not operate?
Length of day, month, or year:
Those are purely kinematic phenomena, and from a purely kinematic point of view the heliocentric and geocentric models are both equally correct within the accuracy limits of astronomical instruments available before the Renaissance.
Sunrises and sunsets:
The sun will rise and the sun will set. We have no idea how the atmosphere works, or how thick it is, because the law of universal gravitation doesn't work in that world. So we don't know if, for example, the sun will appear red at sunset.
Gravity:
Our kind of gravity doesn't work in an universe where the geocentric model is correct. It must be some different force which is called gravity. How it works nobody but you, the author, can say.
Constellations and/or navigation:
No effect whatsoever. The funny thing is that up to this day celestial navigation, as an application of practical astronomy, is done assuming a geocentric model. See celestial sphere for how this works.
Of course, satellite-based navigation systems won't work, because the law of universal gravitation doesn't work.
Any huge effect I don't have enough science to anticipate:
The main huge effect is that only the author can say how that world works, because it most definitely it doesn't work like ours. What keeps water in the ocean, what keeps people on the ground? Does hot air rise? Why? Are there tides? Why?

Note that you do not have to make Sun any smaller or bigger -- whether we adopt a heliocentric or geocentric system has no impact on the distance between the Earth and the Sun.
Everything also applies for the Moon. A Moon may or may not exist; if it exists, it is not universal gravitation which makes it orbit. What is it that keeps the Moon in orbit only the author can decide.

A:

It would be perfectly possible for an extremely advanced civilization, perhaps humans of the future, to create a geocentric solar system.
They could take a rogue Earth-sized planet in interstellar space and create a giant sun satellite orbiting the planet with gigantic fusion power generators generating power for thousands of giant lamps aimed at the planet to heat it and warm it.
If they want the sidereal day of the Earth-sized planet to be similar to that of Earth (23 hours, 56 minutes, 4.0905 seconds) they will have to select an Earth-sized planet in interstellar space that rotates with a similar period and/or slow down or spreed up the rotation of the planet.  If they do that the stars at night will seem to circle with the same speed as on Earth. 
The giant artificial sun satellite will have to orbit at such a distance that the solar day (the time between two successive noons or midnights at the same location) will equal 24 hours.  So that means that the time it takes for the giant artificial sun satellite to make one orbit combined with the time it takes for the planet to rotate once (the sidereal day) will equal 24 hours, a solar day on Earth.  I'm certain there are some users at this site who can easily calculate the distance for you. 
Of course there is the problem that the "moon" should orbit the Earth-sized planet at the same distance that the Moon orbits the Earth in order to have a month of the same length and similar tides.

In Aristotle's (384–322 BC) description of the universe, the Moon marked the boundary between the spheres of the mutable elements (earth, water, air and fire), and the imperishable stars of aether, an influential philosophy that would dominate for centuries.[183] However, in the 2nd century BC, Seleucus of Seleucia correctly theorized that tides were due to the attraction of the Moon, and that their height depends on the Moon's position relative to the Sun.[184] In the same century, Aristarchus computed the size and distance of the Moon from Earth, obtaining a value of about twenty times the radius of Earth for the distance. These figures were greatly improved by Ptolemy (90–168 AD): his values of a mean distance of 59 times Earth's radius and a diameter of 0.292 Earth diameters were close to the correct values of about 60 and 0.273 respectively.[185] Archimedes (287–212 BC) designed a planetarium that could calculate the motions of the Moon and other objects in the Solar System.[186]

https://en.wikipedia.org/wiki/Moon#Before_spaceflight1
So the size and distance of the Moon was measured reasonably accurately about 2,000 years ago.  And a fake moon orbiting a fake earth in an artificial geocentric solar system would have to orbit the fake earth at a similar distance to that of the real Moon.
Which could be a farther distance than than the proper distance for the giant artificial sun satellite to orbit.   Which would be bad because on Earth eclipses are caused by the nearer Moon passing in front of the farther Sun.
There are many other things to consider when designing a possible artificial geocentric solar system.  But presumably some users on this board can do it for you.
A possibly simpler way to create an artificial geocentric solar system would be to find an Earth-sized rogue planet in interstellar space and build a gigantic artificial geodesic spherical structure around it and fit the inner surface of that spherical structure with countless gazillions of lamps.  The lamps would be programmed to turn on and off in patterns to simulate the movements of the Sun, the Moon, the visible planets in the Solar System, and the stars. 
So if it is scientifically possible for an advanced civilization to create an artificial geocentric solar system, a possibly artificial or natural geocentric solar system might exist in a science fiction story set in some parallel universe where the laws of science are different.  And of course a natural geocentric solar system might exist in a fantasy story filled with magic.
As I remember, in J.R.R. Tolkien's legendarium the world was originally not only geocentric but flat, until a great cataclysm where the God of the story changed the Earth into a sphere and made the solar system heliocentric.

