Will the CMB ever stop shining?

Light from recombination is not "constantly shining" and that's why you see it. At a given time in the universe's history (actually a slightly extended period but I'll keep things simple), and only at that time, photons decoupled from the ambient plasma and started travelling freely from all points in the universe. The photon background you see at any time in the universe's history corresponds to the decoupling photons that are just now arriving to you. In other words, at one time, all points in the universe emit a burst of light, and what you see at any given time is the light from just the right distance such that it is just now getting to you. What this means is that there is a constantly receding shell that corresponds to the CMB surface you're looking at. How far away from you that shell is depends on the age of the universe. There's no reason for it to ever stop completely (even if the universe is finite because it will likely have a periodic structure), but it will get fainter over time as the photons get redshifted.

In other words, CMB light will always be present at any time in the universe's history. The horizon of the visible universe can't "pass" the CMB since it's not some individual object. It's the light from an event that occurred everywhere at some early time, and as such you could pick any point in the universe at any time and find a shell around it such that the CMB light from that shell is just now getting to that point. The question "what point emitted light at decoupling that is just now getting to me" always has an answer. The only limitations on observation therefore come from the eternally decreasing energy of the individual photons due to redshifting.

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).

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.)