Photons do not interact with each other, in QED, because there is no such interaction term in the Lagrangian or Hamiltonian. The only interaction term is the electromagnetic field A (4 vector) with charged particles. There are also deep reasons for that. An interaction of two photons would be depicted in QFT as nonlinear in A. The square term would require the photon to have mass, which we know pretty accurately it does not. Other nonlinear terms in A, such as A contracted with some derivative terms cause other problems. The most general form is derived from gauge invariance, and the imposition of other symmetries (like P or T, parity or time reversal) and that the theory be renormalizable (so no infinities), eliminates any other nonlinear terms.
The strong force also obeys symmetries, but the gauge symmetry is non Abelian which allows for the self interactions of gluons. The weak force similarly allows for the self interactions, with some dependence on parity.
The bottom line is that it is not possible to device a spin 1 quantum field (like the photon which is spin 1) that has no mass like the photon, and obeys the symmetries we've observed, and including special relativity, with no infinities, that is self interacting.
The exchange force is a real force, what they mean is that in some cases the photon exchanged between two particles are not seen at the end of the interaction and are labeled virtual photons, as opposed to when we see scattering of charged particles with photons, when you see a photon going in and coming out. They are real forces. In QED and QFT forces are all derived from interactions between particles, and represented by terms that must obey the symmetries we know exist (time, translation, rotation, and in most cases T, C (charge) and P), and special relativity (i.e., the speed of light is the same in all inertial frames of reference). And for electromagnetism an Abelian gauge symmetry which is what causes the field to exist and not interact with itself, and be zero mass. And for all of them no infinities.
When one tries to include gravity quantum theory needs to incorporate General Relativity, and a consistent theory has still to be achieved. String Theory is one contender. So generally one deals with non quantum gravity, with some interactions with quantum fields -- that has been able to be done. I'm just mentioning this to clarify that special relativity is the way relativity fits into quantum theory (quantum theory of fields, QFT), and we'll see what happens in trying to integrate gravity I the future. So, for now, we have a QFT for the 3 forces (electromagnetism, strong, weak), and a non quantum theory for the 4th, gravity, in General Relativity. Each is separately consistent but so far not consistent with each other when gravity is very strong such as in black holes or the Big Bang.