(I'm writing a partial answer to illustrate the kind of answer I'm looking for, although I don't actually know what the answer is. This answer is based on what I figured out after thinking about the question for a bit.)
The principle of relativity states that the laws of physics appear the same to all observers. Note this is, in theory, independent of both special relativity and general relativity; it's not even mathematical.
To make it mathematical, then we need to define what "all observers" means. The first one everyone encounters is the so-called Galilean transformation, which in 1D looks like:
$$x' = x - vt$$
$$t' = t$$
The Galilean transformation is not a physical law; it simply illustrates what we mean by "all observers". We have the $x, t$ coordinates, while the other observer has the $x', t'$ coordinates.
The problem is, of course, under the Galilean transformation, then Maxwell's equations do not remain the same for all observers (i.e., the principle of relativity is violated). Now one could say "therefore Maxwell's equations are wrong", "therefore the principle of relativity is false", or "therefore the Galilean transformation is wrong". The last option is based on the belief that the principle of relativity should hold anyway (I believe historically this was the motivation behind the Lorentz transformations, and the Lorentz transformations were derived before Einstein formulated SR). Experiment would later show that the Galilean transformation is incorrect, and the correct formulation is the Lorentz transformation:
$$x' = \gamma(x - vt)$$
$$t' = \gamma(t - vx/c^2)$$
If you take this to mean "all observers", then Maxwell's equations work.
Now let's consider how this works for quantum mechanics. The most elementary quantum equation is the Schrodinger equation:
$$i\hbar\frac{\partial}{\partial t} \Psi(x,t) = \left [ - \frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)\right ] \Psi(x,t)$$
If we substitute the Galilean transformation into this, we find that this equation remains the same ($\partial^2/\partial x^2 = \partial^2/\partial x'^2$ under the Galilean transformation). Therefore the Schrodinger equation respects the principle of relativity, if you equate the principle of relativity with the Galilean transformation. However we have reason to believe that the Lorentz transformation is the correct transform. Under the Lorentz transform, the Schrodinger equation does not remain the same for all observers.* We can therefore conclude that the Schrodinger equation does not respect the principle of relativity, if we equate the principle of relativity with the Lorentz transformation.
Once again we have several options:
- Perhaps the principle of relativity is false.
- Perhaps the Lorentz transformations is wrong.
- Perhaps the Schrodinger equation needs to be modified.
As far as I'm aware, physicists have generally taken the third option, leading to the Klein-Gorden equation. (One could reasonably attempt to find some other transformation that reduces to the Lorentz transformation in some limit, and also preserves the Schrodinger equation, but for some reason [what?] this is not done). This preserves the principle of relativity, which most people seem to assume must be true. In other words, in standard physics, quantum mechanics respects the principle of relativity.
(Unfortunately I have no idea how this works in GR, so I must end the answer here.)
*This should be obvious - in the Lorentz transform, both $\gamma$ and $v$ include $x$, so the partial derivative does not remain the same either.
