It _will_ happen, but first dark energy needs to be strong enough to expand the universe fast enough (faster than light) so that CMB wouldn't be able to reach anything.
Well there are already portions of space that are expanding faster than the speed of light relative to our position. (see cosmic horizon). The CMB is not just a glowing heat somewhere far away, it's everywhere in the universe in every volume of space.
The moment when every point in space (on a Planck-lenght-scale i guess) will be expanding faster than the speed of light relative to one another, than space-time itself will rip apart and that's the end of our universe - at least that is what the Big-Rip theory proposes.
