Will the CMB ever stop shining?

Classically, the CMB radiation would never completely disappear. It would just become more and more faint and redshifted.

Quantum mechanically, if ΛCDM cosmology is correct, only finitely many CMB photons will reach us, so there will be a last photon.

(edit: In general big bang cosmologies, there may be no last photon. I formerly said that the photon density is finite and that means there are finitely many photons per astronomical body, but ticster pointed out in a comment that there's no guaranteed last photon for any particular detector. In fact, I suppose in principle every astronomical body could receive an infinite number of CMB photons, infinities being what they are.

Also, it might be worth pointing out that CMB light mixes with light from other sources and photons are indistinguishable particles, so this question is not strictly meaningful unless the CMB is the only source of light in the universe.)

https://en.wikipedia.org/wiki/Chronology_of_the_universe

"The spherical volume of space which will become the observable universe is 42 million light-years in radius at this time."

So essentially it is CMB light from every point from this "then size" of the Universe is reaching us and we happen to see at present or at any time CMB light from all such points within this universe that are at just the distance from you. But to my mind this is limited to the perimeter of the then Universe which was all of 42 million light-years in radius.

If the Universe were "just expanding", there would be no limit to amount of CMB photons received. This is the gist of the Ant on a rubber rope puzzle: No matter how fast you expand a rubber rope / a Universe, it is always possible to travel an arbitrarily large distance in a finite time. And since CMB photons were emitted from everywhere in the Universe almost at once, they'd keep coming.

However, due to the accelerated expansion, there is a limit. If the acceleration is indeed how we think it is (due to a non-changing cosmological constant), the observable Universe asymptotically increases to a finite size in comoving coordinates. That is, the physical size will increases without limit, but the part from which we may receive information contains a finite number of galaxies, atoms, and photons.

This part — the maximum (comoving) size of the observable Universe — is today 63 billion lightyears (Glyr) in radius, whereas the Observable Universe today is only 46 Glyr in radius. Hence, we have already received $mathbf{sim(46/63)^3 = 40%}$ of the CMB photons that we will ever be able see.

Thus, the CMB photons will become more and more redshifted, and arrive and an increasingly slower rate, until one sad and gray autumn day, our ancestors will see the very last CMB photon (which will have redshifted so far into the radio regime that it can hardly be called a photon anymore).

I am answering with respect to a finite universe only. There will come a time when the radius $R_{ou}$ of the observable universe will be $$R_{ou} = pi R_{curv}.$$ This is as far as you can see any emitted photons. If $$R_{ou} < t_{CMB}$$ where $t_{CMB}$ is the time it would take for a CMB photon to reach the observer in a non-expanding universe, then the observer will not be able to see such photons.