I'm actually rather surprised none of the other existing answers (at time of writing) mention shot noise. So, I'd like to talk a little bit about this, but since there is also a bounty asking specifically about the how quantum field theory plays with voltages, I'll have a couple of things to say about that as well.
There are many forms of noise in electronics. For example there is (approximately) white thermal noise (also called Johnson-Nyquist noise which is the voltage fluctuations caused by the thermal motion of electrons in resistors throughout the device. There's also $1/f$ noise which is often called pink or flicker noise since it was originally observed in vacuum tubes (a low-frequency visual flickering), though I believe the origins of flicker noise are still not very well understood.
Shot noise is another type of noise which is caused by the discretization of electric charges into electrons. Basically the idea is that as current flows through a junction, like a diode, vacuum tube, or anything else really. The effect is, in most applications, smaller than the other types of noise I mentioned, so it sometimes gets overlooked, but it's there nonetheless. The idea is that the current flowing through a junction tells us the mean number of electrons per second passing through, but the number of electrons which actually pass through the junction in a fixed interval of time is a Poisson process, and so there are some fluctuations about the mean in the number of electrons, and then those fluctuations imply fluctuations in the current (and hence voltage) in the circuit.
Since we have discrete numbers of electrons jumping across our junction, we will also have (approximately) discrete jumps in the current (and hence voltage) to do with it, which is why shot noise came to mind when I saw this question. All of this, of course, is really just a semi-classical approximation making use only of the discretization of charge into electrons and does not represent a fundamental discretization of the potential.
So then, let me comment on how quantum field theory (QFT) plays into this since the bounty asks about it. The short answer, which might be a little disappointing, is that even in QFT there is no discretization to the potential, but let me explain a little rather than just leave it there.
The name and language surrounding QFT would seem to imply that fields and everything else are "quantized" in the sense of being "discretized," but these are actually distinct things: Just because something is quantized does not mean it is discretized. So let's just look at what quantization, at least roughly, means. As a disclaimer, everything I say here is going to be a bit informal. To get a full picture of what's going on one would really need to know a little QFT, to which many books are devoted.
In a classical (field) theory, we have a bunch of fields and we specify the "state" of the system by specifying a particular field configuration. So for example, in electrodynamics our fields are the potential $V$ and the vector potential $\vec A$, which we usually put together into a 4-vector $A_\mu$ which we refer to as "the" vector potential. To specify what's going on, we would just need to specify a vector potential which obeys Maxwell's equations.
In a quantum (field) theory, we "promote" all our fields to operators and to specify the state of our system, we specify a vector in a Hilbert space (a vector space with a notion of inner product and some other technical niceties) instead of specifying a field configuration which obeys the equations of motion. So for example, in electrodynamics we would now have 4 operators $\hat A_\mu$ which act on vectors in our Hilbert space.
Ignoring a bunch of subtleties which aren't important to the idea I want to get at, we can actually do something in QFT which makes things more similar to classical mechanics. See, since the Hilbert space is a vector space, we are free to choose a basis of vectors however we like. The "standard" and "nice" way to choose a basis is to use the eigenvectors of some operators, such as $\hat A_\mu$.
There are some technical issues with this involving gauge invariance, but here's the picture we should have in mind. Imagine specifying some field configuration everywhere in space and time, and then just say $|A(everywhere)\rangle$ is the state which corresponds to this configuration we have imagined. Then, by definition, $\hat A_\mu(x)|A(everywhere)\rangle = A_\mu(x)|A(everywhere)\rangle$ where $\hat A_\mu(x)$ is the vector potential operator at the location $x$ in space and time while $A_\mu(x)$ is the value of the vector potential at $x$ which we have imagined.
So, whereas in classical mechanics we could specify the state of the system by a field configuration which obeyed Maxwell's equations, in QFT we can specify the state of the system by (a linear combination of) field configurations, which may or may not obey Maxwell's equations.
This is the main idea I wanted to get at: even in QFT we can still import some of our thinking about field configurations from classical mechanics, just with a few new bells and whistles. More to the point, at no stage here do we encounter a notion of quantization of the field configuration itself. So, while some things are quantized in QFT, not everything is, and in particular the electric potential is not. At least, not any more than it is classically (which is approximately is in the case of shot noise I mentioned earlier).