What Willie's answer shows is that, for some non-trivial Lorentz-translatable cellular automaton, every cell would need an infinite number of neighbors. This cellular automaton couldn't work if any cell had an infinite number of live neighbors. As was pointed out in the comments (2018), it could work if there were only a finite number of live neighbors of any cell. One would need to impose constraints on which kinds of configurations are allowed to ensure this, though.

There's a way of getting around this, however. You could make each cell correspond to a point in space-time and also a boost (a boost is essentially a velocity in the Lorentz group). Then, cells would interact with cells both close to them in space-time and also close in boost. I don't know whether anybody has considered cellular automata like this.

In order for this to have a correspondence to realistic quantum field theories, it would have to be the case that when two particles interact at a high boost, the interaction strength goes to 0 as the boost goes to infinity. I don't know whether this is true, although the thought experiment of considering particles falling into a black hole through a sea of Hawking radiation makes it seem like it might be.

doubledthe number of google hits to "relativistic cellular automata"? :) $\endgroup$2more comments