Anything CAN happen, but most rabbits are able to live outside without dying from hearing predators.
I used to raise rabbits for meat. I only ever had one rabbit dies from something like that. Dogs got inside a fence and were tearing open cages and killing rabbits. Some rabbits were killed by the dogs, and one was untouched but dead in the cage (presumed heart attack), but at least a dozen rabbits were alive (traumatized, but alive.) Dogs were seen outside the fence on other occasions, barking and scratching but unable to get in, and no rabbits ever died from that (over several years' time, this amounted to hundreds of rabbits that did NOT have heart attacks from any cause, even when predators were close enough to hear, smell, and see.)
I would be much more worried about whether predators can get into the cage and kill the rabbits, than about whether the rabbits would die from just hearing predators that are unable to get in.
That is definitely a consideration. Different climate can make a big difference, and I don't see any mention of where OP lives. Shade and plenty of cool water are enough in some climates, but not in some other climates.