In article , Martin Brown
writes:

I suppose if the universe was of finite extent the CMBR could go out
suddenly tomorrow. It does beg an interesting question in the very long
term future

I have to point out that it merely asks the question. Begging the
question means essentially answering a question with a statement which
repeats the question, e.g. "Why did you come here?" "Because I had a
desire to do so."

- will the CMBR gradually fade out to T=0 and time dilate as
the surface of last scattering approaches the particle horizon or will
the dark energy acceleration make it switch off sharply?

The particle horizon always increases in comoving coordinates and, in an
expanding universe, in proper distance as well. It is always behind the
surface of last scattering. The redshifts of both increase with time,
but it doesn't make sense to say that the latter approaches the particle
horizon.

If the cosmological constant comes to dominate the expansion, which
appears will happen in our universe, then the universe will
asymptotically approach the de Sitter model. This model has an event
horizon at a fixed proper distance. As the universe expands, objects
approach it, becoming infinitely redshifted when reaching it. There is
no sharp switch-off.