Accelerating an electron to a high speed

Suppose there are two vertical metal plates. They are separated apart by a small distance. While one is grounded, the other one has a potential of some $V$. Now suppose electrons are produced at the grounded plate. Since there is a potential difference the electron will be accelerated to the plate with some final speed:

$$v=\sqrt{\frac{2qV}{m}}$$ where $q$ is the charge of an electron and $m$ being the mass of an electron. The voltage and the speed are ~proportional to each other, and it seems there is no limit to how fast it can go.

Now, it is known that of course there is a limit, namely the speed of light. What would happen if I turn the voltage all up to 256kV?

When the voltage is high enough so that the speed becomes relativistic, two effects precludes becoming superluminal. First you must now use the relativistic form of newton's second law, with 4-vector velocity and acceleration. Second you must keep in mind that an accelerating charged particle would radiate with a power proportional to the square of its acceleration (for non relativistic speeds), and proportional to $a^2\gamma^6$ for the relativistic case in which acceleration and velocity are collinear (such as the example you proposed)