My old theoretical physics professor used to say that (already) (special?) relativity shows that "the future is not predictable". Any ideas how this should be interpreted?
Unpredictability and special relativity can come from the fact that objects that are at a space like separation from us can influence our future like cone.
For example, if Alpha Centauri exploded in a supernova right "now" in our reference frame, we would not know about it for 4 years since that star is 4 light years away from us. So we can not now predict that 4 years from now we will be hit with the supernova blast wave.
As @Arnoques succintcly put in his comment, to predict an event in our future light cone, the entire past light cone of that event would need to be known and that would include events that are outside our present light cone.
The "(already)" in brackets in the question suggests that the statement is about classical (non quantum) predictability. Given that, I would guess that the unpredictability your prof is talking about is the unpredictability that comes along with general relativity due to the singularity theorems.
These theorems hold provided certain "reasonable" conditions apply. There are lots of technicalities in setting up the statement of the singularity theorems, but basically you end up proving that certain geodesics (the world lines followed by particles in free fall under gravity) cannot be extended indefinitely. Effectively, spacetime just “ends” for particles following these incomplete geodesics hence predictability breaks down.