UV completion means describing the degrees of freedom and their dynamics at high energies, i.e. short distances. QCD is an example of such a theory: the quarks and gluons are the degrees of freedom that describe the dynamics of the strong interaction at short distances as a local quantum field theory, that can be treated in perturbation theory, since asymptotic freedom implies that the coupling constant decreases with incrreasing energy and quantum corrections are under control from the renormalization group.

A theory is IR free means that it describes non-ineracting degrees of freedom in the ``infra-red'', i.e. at low energies, large distances. QCD is *not* IR free: the quarks and gluons interact strongly at large distances and become baryons and mesons. How this can be described quantitatively is the subject of lattice QCD and a major research program.

A theory that is IR free is QED, where the coupling decreases with distance and, asymptotically, electrically charged particles are free at infinity.

UV completion means describing the degrees of freedom and their dynamics at high energies, i.e. short distances. QCD is an example of such a theory: the quarks and gluons are the degrees of freedom that describe the dynamics of the strong interaction at short distances as a local quantum field theory, that can be treated in perturbation theory, since asymptotic freedom implies that the coupling constant decreases with incrreasing energy and quantum corrections are under control from the renormalization group.

A theory is IR free means that it describes non-ineracting degrees of freedom in the ``infra-red'', i.e. at low energies, large distances. QCD is *not* IR free: the quarks and gluons interact strongly at large distances and become baryons and mesons. How this can be described quantitatively is the subject of lattice QCD and a major research program.

A theory that is IR free is QED, where the coupling decreases with distance and, asymptotically, electrically charged particles are free at infinity.

Yes there are connections with QCD. In any quantum field theory amplitudes are expressed as infinite sums. One has to check whether after taking into account contributions from all possible momenta, infinite sums are convergent or not. Divergences which we face are of two types. 1. Infrared divergences and 2. Ultraviolet divergences.

Infrared divergences arise from the propagator, whereas ultraviolet divergences appear from the upper limit of integration. Ultraviolet divergences are removed by regularization and renormalization. If it is not possible to remove ultraviolet divergences, then the theory is not perturbatively well defined at very short distances. As Stam has pointed out QCD is an example of a renormalizable theory, which means that ultraviolet divergences are under control.

See: http://en.wikipedia.org/wiki/Ultraviolet_divergence

In QCD theory presence of mass-less gluons produce divergences for low momenta. So this theory is not IR free. Again in QCD theory when you increase momenta to very high values then new divergences will appear which can be removed by regularization and renormalization. Therefore in principle QCD can be a valid theory all the way up to the Planck scale. Nonetheless,  if a new symmetry such as the GUT symmetry takes over then that will be the UV completion of QCD theory. This is because a new theory is taking over from the GUT scale and QCD is a part of that bigger theory. GUT scale becomes a natural cut-off scale in this case. This will reduce gauge hierarchy problem.

Another example of UV completion is very well known. We know that the four fermi interaction theory of weak interactions is non-renormalizable. But SU(2)LxU(1)Y theory is renormalizable. In that sense the latter in the UV completion of the former.

See for example: Book by D. Bailin and A. Love on Gauge field theory.

No new divergences appear in QCD, since it's a renormalizable theory. Furthermore, since it's asymptotically free, it doesn't have a Landau pole problem.

Yes I have edited my previous answer and included your point.

Infrared divergences, also, are removed by regularization and renormalization. However these issues are independent of the notions of UV completion and IR freedom.