Conventionally, we talk about gluons as having a zero rest mass, but that analysis is model dependent (the model assumes a constant mass subject only to Lorentz adjustments) and based on a fit to exclusively UV range data.
The emerging consensus is that faster moving gluons approach zero mass, while slower moving gluons approach a finite mass on the order of 600 MeV +/- about 15%.
The interpretation of the Landau gauge lattice gluon propagator as a massive-type bosonic propagator is investigated. Three different scenarios are discussed: (i) an infrared constant gluon mass; (ii) an ultraviolet constant gluon mass; (iii) a momentum-dependent mass. We find that the infrared data can be associated with a massive propagator up to momenta ~500 MeV, with a constant gluon mass of 723(11) MeV, if one excludes the zero momentum gluon propagator from the analysis, or 648(7) MeV, if the zero momentum gluon propagator is included in the data sets. The ultraviolet lattice data are not compatible with a massive-type propagator with a constant mass.
The scenario of a momentum-dependent gluon mass gives a decreasing mass with the momentum, which vanishes in the deep ultraviolet region. Furthermore, we show that the functional forms used to describe the decoupling-like solution of the Dyson–Schwinger equations are compatible with the lattice data with similar mass scales.
From O Oliveira and P Bicudo, Running gluon mass from a Landau gauge lattice QCD propagator (2011) J. Phys. G: Nucl. Part. Phys. 38 045003 doi:10.1088/0954-3899/38/4/045003.
At any rate, that is my take on what to make from analysis of the current state of research that is often stated in terms that make less direct statements about quantities and properties of QCD entities familiar to non-QCD specialists than the abstract from a 2011 paper cited above.
This "decoupling solution" to the QCD equations appears to yield a stable result which should be confirmed by physical results if they can ever be made, while the alternative, a "scaling solution," in which a zero momentum gluon is massless, is apparently unstable and hence unlikely to have a physical manifestation.
The implications of the decoupling solution are weird. In special relativity, a massive particle gets gains effective mass as it goes faster. In the QCD decoupling solution, a massive particle loses effective mass as it goes faster. Special relativity "feels" a bit like air resistance. QCD "feels" a bit like the relationship betweeen the friction experience by something moving along a snow covered surface and its speed.
But, there are elements of a massive gluon approach that are quite attractive and intuitive in light of the other data and constraints. It helps explain why the mass of ordinary protons and neutrons and also more exotic combinations of quarks seems to come mostly from the gluons and not the quarks. It fits the intuition that massive bosons should be associated with short range forces, while massless bosons should be associated with long range forces. It fits the notion that the strong force (which gets more insurmountable with distance and nearly vanishes at short range) looks a bit like the inverse of gravity (which gets weak at long range and strong at shorter ranges). It provides a way to reconcile disparate data from low energy and high energy contexts that appear to be inconsistent with a constant gluon mass or a massless gluon.
Not every little glitch in the theory has been neatly ironed out and presented definitively, but the trends towards this interpretation becoming dominant seems to be mounting.