Other answers have covered the key points correctly, but I'll jump in too, maybe emphasizing a slightly different angle.
It's not just that energy is not conserved -- even defining the total energy of the Universe (or even the total energy in any reasonably large volume) is problematic and, in some sense, unnatural.
What people usually have in mind when they talk about the total energy of the Universe (or a large volume -- from now on I'll stop writing that) is something like the following: Figure out the energy of each particle in that volume, and add 'em up. That's a sensible procedure for figuring out total energy in other contexts: it works great if you want to talk about the energy in all of the air molecules in this room. But it only works if all of the individual energies are determined in the same inertial reference frame. And in the expanding Universe (or any curved spacetime), there are no inertial reference frames that cover the whole region.
When people worry about energy non-conservation as applied to CMB photons, what they're implicitly doing is calculating each photon's energy in the local comoving reference frame (the one that's "at rest" with respect to the expansion). But all of the different comoving frames are in motion with respect to each other, so it's "illegal" to add up those energies and call the result a total energy.
Think of a Newtonian analogy: if one person measures the kinetic energy of something on board a moving airplane, and another person measures the kinetic energy of a different object on the ground, you can't add them up to get a total energy. And certainly the sum of those two things won't be a conserved quantity.
Just to be clear: I know that there are a bunch of contexts (e.g., asymptotically flat spacetimes) in which it does make sense to talk about energy conservation in various forms. But in this specific context, I think that the above is the essence of the issue.