First of all, note that the news article is from October 2013 and so it is somewhat outdated. However the experiment was actually scheduled to start in summer 2016 at the Jefferson Lab, and so the results are yet to come.
I took a quick look at the paper linked in the news article and the words "massive photon" do not appear anywhere. Instead they refer to the search of a "new boson" at some point. So it is quite likely that the expression quoted in your link is just an easy way to give the journalists a rough idea of what kind of particle these scientists are looking for. It is not related in any way with the photons we all know, but it just shares with them some properties.
The particles with integer spin are called bosons, because they follow the Bose-Einstein statistics. Photons are bosons, the Higgs boson is a boson (!), and the new particle they are looking for is supposed to be a boson. All the known bosons are included in the Standard Model and, with the exception of the Higgs, play the role of carriers of three of the fundamental forces (electromagnetic, weak and strong). The new particle that they hope to find with this experiment is not predicted by the Standard Model, and therefore its detection would lead to "new" physics. It is quite common that, within several different theoretical frameworks, new particles are conjectured in order to explain the origin of dark matter, but so far no attempt has proved successful. We'll see how this goes!
Back to the actual question: the experiment is aimed at finding a new massive gauge boson, called $A'$ in the literature, with mass in the range $0.01-10$ GeV. One theoretical model that describes how this new dark matter candidate could arise is explained in detail in this paper, and a more readable summary of how this could relate to current astrophysical observations can be found in section A of the research proposal for the experiment. I suggest to check these two resources for detailed information on the subject. The general idea behind this theory is that the dark matter itself is constituted by very massive ($\sim$TeV) particles, which are therefore out of the reach of experiments (The current LHC energy might actually be borderline, but no results have been obtained so far). So far, this is no different from many other theories of dark matter, including the ones that want to identify the new particles with the supersymmetric partners of known particles. The difference, however, is that while the most common idea is that these massive particles interact through the weak force -the so called WIMPS- in this case the interaction takes place due to a new "fifth force" mediated by a new massive $U(1)$ gauge boson, the "Dark Photon".
According to this theory the $A'$ is kinetically mixed to the Standard Model $U(1)_Y$, which means that in the Lagrangian we have a term of the form
$$\mathcal{L}_{mix}=\frac{\kappa}{2}V^{\mu\nu}B_{\mu\nu}$$ where $\kappa$ is a parameter that determines the amount of mixing and $V^{\mu\nu}$ and $B^{\mu\nu}$ are the field strength built from the "Dark" and "normal" photon fields respectively. In short, this means that the "dark photon" interacts with matter in a way which is very similar to the traditional photon. In particular it implies that it can decay into electrons, muons and pions and this could possibly justify some excesses in the production of these particles revealed by previous astrophysical observations.
One great hope of the scientists was that this could provide a solution for the anomalous magnetic moment of the muon puzzle, but unfortunately this possibility has been largely ruled out by other experiments, see for example this. To conclude, it seems that the presence of a "dark photon" would not be inconsistent with other observation, but this would require a certain amount of fine-tuning of the parameters of the model and fails to explain one of the phenomena that was at the basis of its proposal.
