All particles exhibit wave-particle duality and interference effects, not just photons. However, it would be difficult to observe this aspect of dark matter because of its extremely limited interactions with our own kind of matter.
For example, in the CERN experiment you mention, they are simply looking for "missing energy" - less energy in the products of a collision, than went into it. In some of the theories being tested, there are messenger particles that interact with our Higgs and with dark matter, and if the Higgs produces a messenger that decays into dark matter rather than ordinary matter, then the energy of the messenger is lost to detection, thus "missing energy".
On the other hand, to observe interference effects in such a theory, you might need e.g. a collision between two Higgses, each of which emits a messenger, and then the messengers exchange a dark photon. This would be a very rare event, so hard to detect and analyze... Since dark photons by definition interact with dark matter, your best chance of seeing them interfere, is if you yourself, and your sense organs or experimental apparatus, are made of dark matter.
There are a few ways in which the quantum nature of dark matter might show up. If there are "dark atoms", they may emit dark photons at specific wavelengths, and this might eventually, indirectly, have a manifestation in the world of ordinary matter, like the distribution of cosmic rays from the galactic center, an environment in which extreme interactions would occur in great quantity.
Alternatively, there are theories of dark matter in which the halo of dark matter surrounding a galaxy, rather than being like a big cloud of dust, exhibits some quantum properties on astronomical scales, like Pauli exclusion or superfluidity. So there might be traces of this on a vast scale, in the distribution of ordinary matter (stars, gas).
edit: There's a paper today on "Dark Matter Interferometry".
