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: 
There is a simple argument why photons emitted by stars can't be dark matter, and that's because there is about ten times more dark matter than normal matter. If all the stars created at the Big Bang had turned into photons there still wouldn't be enough of them.

You might argue that maybe more normal matter than we think was created during the Big Bang, but the theory of [Big Bang Nucleosynthesis](http://en.wikipedia.org/wiki/Big_Bang_nucleosynthesis) places a [limit](http://cosmology.berkeley.edu/Education/CosmologyEssays/BBN.html) on how much normal matter was created, and this limit is four times smaller than the amount of dark matter. The dark matter has to be something *odd*.

If you're interested in more info [this paper](http://arxiv.org/abs/0906.2087) is a good review, though harder going than the answers here!
I just want to point out that it seems some people may be conflating 'dark matter' with 'dark energy.' Regular "normal" matter like electrons, neutrons, and the like, are estimated to make up about 5% of the matter/energy density of our universe.

Dark matter, estimated to make up about 25% of the matter/energy density of the universe, is matter that has mass, but the gravitational and other effects of which are not directly visible. Dark matter is somewhat mysterious but could easily be something like exotic particles or oceans of black holes between galaxies. 

Dark energy, is the real mystery; it makes up the other (about) 70% of the matter/energy density of the universe needed to explain inflationary cosmology and expansion/acceleration of the universe.

Photons are mass-less particles that embody energy, visible when they strike objects. I think it's an interesting notion that the energy from photons could at least in part constitute some of solution to the "missing" dark energy problem. As has been pointed out, it's difficult to reconcile how so much of the "missing" energy could have come from so little: the 5% ordinary matter creating all that dark energy. But I don't see anything impossible about this idea generally. Perhaps the dark matter also contributes to this somehow. There may even be dark-electromagnetic forces that create dark-photons, this may be seen as extra-dimensional as one poster referenced earlier. 

"It’s humbling to think that ordinary matter, including all of the elementary particles we’ve ever detected in laboratory experiments, only makes up about 5% of the energy density of the universe." \_Sean Carroll

With so little of what we are used to seeing and interacting with in ordinary meaningful way actually making up what exists, speculation of what else is out there is not only justified but necessary.