Why doesn't infrared slavery provide a proof for quark confinement? It is well known how and why the running coupling of QCD grows as we reduce energy or equivalently increase distance. Hence, if the effective color charge increases with distance then why doesn't this infrared slavery fully explain quark confinement?
 A: Confinement is not about the running coupling. By confinement we mean a qualitative change - a phase transition - in how the physics between two quarks works compared to e.g. two electrons.
In particular, confinement is the phenomenon that, in the confining phase, the force between two quark rises linearly with the distance between them, or at least that the energy required to separate two quarks is infinite. No amount of running coupling can explain this, since this is not related to the value of a coupling at all, but to the the functional form of the force law/free energy functional.
To see that a running coupling cannot explain this, consider the traditional force law in QED (i.e. Coulomb's law), then crank up the coupling. No matter how high the coupling goes, the energy required to separate two oppositely charged particles is finite.
The traditional lattice treatment (see also this answer of mine) identifies the Polyakov loop's expectation value as the order parameter of the confining/deconfining phase transition, and the Polyakov loop is essentially (and a bit handwavingly) the exponential of $-E$, where $E$ is the energy required to separate a two-quark system. As long as the Polyakov loop is zero, $E$ is infinite, i.e. the quarks are confined.
A: (I am reproducing and elaborating on my comments above here as an Answer at the request of @Diracology, and take the opportunity to clarify some other comments.)
We only know about that running of the QCD coupling in perturbation theory. It confirms the idea of asymptotic freedom which early experiments had indicated, and for which Noble prizes were awarded. That such a running is a physical phenomenon is summarised by data in this paper. 
However, we do not know if the coupling will keep getting stronger at low energies (long distances) beyond the perturbative domain in which we can do reliable calculations.  If indeed the effective coupling keeps getting stronger, it is possible that strong interaction dynamics might lead to a dramatic change (phase transition) to a novel state of the system. 
So to study the strong coupling domain we need non-perturbative tools, such as lattice gauge theory, or some toy models. 
Indeed, lattice gauge theory indicates that QCD has a confining phase, characterised by a linear quark potential similar to what one finds in lower dimensional analytic models. 
One physical picture for quark confinement is the t'Hooft-Mandelstam "monopole condesation" leading to a "dual superconductor" whereby colour electric flux forms strings between charges, leading to a linear confining potential (Recall that in the usual Meissner effect in superconductors the condensation of electric charges (Cooper pairs) leads to magnetic flux tubes). 
Seiberg and Witten illustrated the monopole codensation idea for confinement in a SUSY model of QCD, adding support to a similar idea in usual QCD.
In summary, confinement in QCD seems to be associated with a different phase from that which we associate with the asymptotic freedom perturbative calculations. But it is unclear (to me at least) how much of a role the strong coupling dynamics plays in the transition to the confinement phase.
**As an aside, there are many ways of interpreting and calculating the beta function beyond what is done in textbooks which usually discuss it in the context of renormalisation (usual or Wilsonian). The running of the beta function with energy scale can be physically interpreted in position space (and also calculated, if you like. I am sure someone must have done it) as the effective interaction between a source charge and a probe charge, just like in Coulomb's law. The vacuum is polarised by virtual particles in QFT and so the effective interaction changes. 
Indeed in this picture its easy to see that the effective charge will depend on the distance between the source and probe. So the beta function can also be interpreted as telling us how the effective charges vary with distance between the charges. 
If you want to amuse yourself, you can also calculate the same beta function by looking at the interaction of the charge with an external magnetic field, which provides the clearest physical picture of why spin one is different from spin 1/2 or spin 0. See this paper by Nielsen.
The basic reason why all the different methods give the same beta function is that deep down it physically corresponds to the effective interaction between charges. It does not matter whether you calculate it in momentum space or position space or between a charge and a current carrying wire, or between a charge and a external field etc.
