Question: After a star lives and dies, I assume virtually all of its mass would be photons. If enough stars have already lived and died, couldn’t there be enough photon energy out there to account for all the "missing mass" (=dark matter) in the universe?

And if there were enough photons to account for all the missing mass, what would it look like to us?

Answer: 
Photons are easily detectable. We can count how many photons are there at any distance of us by just counting the photons reaching us from there. It is impossible that the hidden photons ramble the whole universe but mysteriously avoid us.
Photons don't have mass. So your assumption's incorrect, although I don't know how much of a (say) main sequence star's mass gets converted into photons over its lifespan. 

Photon's can't account for, say, dark matter, because *dark matter has mass*.

Thermal energy (in the vacuum) is comprised of photons, which can spontaneously form particle-antiparticle pairs. Usually these quickly annihilate, so this is also not a good source of mass.