While proving that no matter can reach the speed of light (a fact which I call the kinetic energy barrier), Einstein uses the fact that he can calculate the velocity and position of an electron. However, if quantum effects apply, then it seems to create a problem in Einstein's assumptions themselves. How is the proof of the kinetic energy barrier true even though quantum effects exist in nature?
It is well established by now that the speed of light barrier does not apply to quantum particles, and this property makes the construction of relativistic quantum field theories and other relativistic quantum systems tightly constrained. The argument that you can't transmit signals faster than light is fine, but particles are not necessarily signals, because if you make a particle over here, and measure a particle over there, you might not be measuring the particle you created, but another identical particle you created from the vacuum.
So relativistic field theory, with its faster-than-light particles, requires that there are no unique particles, that all particles have identical copies. Further, the faster than light motion can be back-in-time in different frames, and back-in-time motion means that every particle must have a back-in-time partner, called its antiparticle.
The quantum field restores locality. So even though quantum particles can propagate faster than light, information can't propagate faster than light. The quantum fields are the quantities which tell you what information you can gain locally by experiments. The particles in a Hamiltonian formulation are nonlocally related to the quantum fields (but the two are related more simply in a particle path integral formulation).
The proof that quantum particles cannot be restricted to less than light speed is simple: the restriction that the energy is positive means that the frequency is positive, while the restriction to forward propagation inside the light cone means that the propagator vanishes outside the future lightcone, so in particular, into the past. and there is no function which vanishes in a half-plane whose Fourier transform also vanishes in a big unbounded region like this. This is covered in this answer: the causality and the anti-particles
Light speed being invariant is necessary for there to be "quantum effects" (if by that, you mean probabilistic phenomena).
Since light speed is invariant, fixed at a rate of 1 lp per 1 tp, and no measurements are possible below the Planck scale, therefore any speed slower than c must be described as a probability at that scale.
Even if you plot the position of a wavefront as smoothly continuous regardless of scale, when you superimpose the smooth path of that wavefront on graph paper taking each square to be 1 lp by 1 tp, since the slope of the path is the velocity, at speeds below c there will increasingly be occurrences where two or more consecutive steps forward in time may fall between two consecutive steps forward in distance.
Alice, without knowing the initial state of the system, upon observing the particle at a certain point, even if she knew its velocity, would never be able to know for certain where the particle would be at the next Planck time, because there would always be a probability less than 100% that particle would move 1 lp relative to her frame of reference.
Considering that, it seems that the invariance of c is indeed what forces speeds slower than c to be expressed as probabilities at the scale below which observation cannot theoretically distinguish differences. Perhaps this seems like a circular argument, but, consider that if the maximum possible speed (c) could change, then the Planck length would change too.
Regarding the other answer that says quantum particles can go faster than light speed, I'm not sure how useful that answer is, even if it's true, because a "quantum particle" that moves faster than light is not in the same category of things as everything else we think of as a "particle." Furthermore in light of the work of Joy Christian and John Bush, et al., there does seem to be some debate on whether the interpretation of QM accurately represents what is actually going on behind the scenes. Although there's no doubt as to QM's predictive accuracy, the cosmological picture it paints is by no means scientifically proven the same way that the speed of light is.
If we limit ourselves to only a discussion about what is directly observable and what actually constitutes "information," I think we can avoid some semantic confusion. When we say "speed barrier" we are referring to a velocity that an observable or real object can travel at, and we should leave out of the discussion any imaginary or virtual things that cancel out of the equation by the time any effect can be observed.
Correct me if I'm wrong, please.
Because no one's figured out a way to use and/or validate it, but see Has anyone tried transmitting a signal with equally-tuned Casimir plates across the quantum vacuum?.