Will we ever reach a point where all the light that was set free by recombination finally reaches us from our point of view and the CMB stops "shining" for observers on Earth?
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.
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.)