Majorana particles have 0 mass. Dirac equation decays into 2 Majorana equations if Dirac mass is 0.

How can we check if the Majorana mass term is Lorentz invariant or not?

The difference between Dirac and Majorana fields is that Dirac fields are general spinors while Majorana fields are the specific case of Dirac fields for which self-charge conjugation has been imposed: as a consequence of the fact that in QFT the field's complex character is connected to its charge, then Majorana fields have to be invariant by charge inversion and this makes them electrically neutral, which is why only the neutrino can be a Majorana field. This is also related to the fact that Majorana fields have to be their own antiparticles, and as such you do not need to have a Majorana spinor and a Majorana anti-spinor to annihilate because two Majorana spinors would annihilate just as well, with the consequence that if they existed we should be able to observe a neutrinoless double-beta decay.

Finally, for the Majorana mass term the thing to notice is that you need to employ Grassmann variables if you want it to be different from zero, but once this is settled then the actual proof that this non-zero Majorana mass term is Lorentz invariant is done exactly like the proof that the Dirac mass term is Lorentz invariant.

Hope this helps.

Photon is also a Majorana particle. However, under Lorentz transformations photon transforms as a vector not spinor.

Regarding the neutrino, it could be either a Dirac fermion or a Majorana fermion. If it is a Majorana fermion then one expects that a Majorana mass term should be allowed which violates Lepton number by two units. Observation of neutrinoless double beta decay will establish that neutrino is a Majorana particle.

http://pdg.lbl.gov/2015/reviews/rpp2015-rev-neutrinoless-double-beta-decay.pdf

... mmh, well, to be precise a Majorana particle is not a particle that coincides with its antiparticle but a fermion that coincides with its antifermion... so although the photon is its own antiparticle nevertheless it is not a Majorana particle.

The spinor field representing the neutrino cannot be equivalent to that of the antiparticle. A method for differentiating between

the neutrino and the antineutrino within the transformation rules of quantum field theory may follow from the postulate of a

new spinor with a variation of the definition through the conventional charge conjugation matrix.