Why doesn't the color force between two quarks have an inverse square law? From what I understand the color force between two quarks doesn't decrease with distance. Why is it that the color force doesn't decrease with the square of the distance if there are three dimensions of space?
 A: Force is a classical concept.  Even classically not all forces decrease with distance, as the forces acting on a spring show,  which increase with distance.
I do not see the connection with the three dimensions of space and the functional form of the potential of a force. The strong force is modeled with the exchange of gluons which are depicted  in Feynman diagrams as little springs, exactly because the potential acts like the spring potential.

Inside a baryon, however, the color force has some extraordinary properties not seen in the strong interaction between nucleons. The color force does not drop off with distance and is responsible for the confinement of quarks. The color force involves the exhange of gluons and is so strong that the quark-antiquark pair production energy is reached before quarks can be separated. Another property of the color force is that it appears to exert little force at short distances so that the quarks are like free particles within the confining boundary of the color force and only experience the strong confining force when they begin to get too far apart. The term "asymptotic freedom" is sometimes invoked to describe this behavior of the gluon interaction between quarks. 

So it is not simple.
A: This is covered by a few existing answers (see for example About free quarks and confinement) though surprisingly it doesn't appear that anyone has asked this exact question before.
Anyhow, the answer is that the colour force is mediated by particles called gluons just as the electromagnetic force is mediated by photons. The difference is that while photons are uncharged, gluons carry a colour charge so they themselves also feel the colour force. In effect the lines of the colour force interact with each other so they don't spread out evenly.
In the electromagnetic field the lines of force radiate outwards symmetrically from an electric charge, so at some distance $r$ they end up spread evenly over a spherical surface of area $4\pi r^2$ and this gives us the inverse square law. When two particles with a colour charge interact, the lines of the colour force don't spread out evenly. Because the lines of the colour force interact with each other as well as with the particles at long distances they form into a flux tube (also known as a QCD string) between the particles. Because the lines of force are not evenly spread out over a sphere the force doesn't fall off as $r^{-2}$.
