I don't think they die, even if 100% replaced. However, I do not think they are made immortal either.

Let's not kid ourselves. Any system capable of replacing the human brain will need to be technologically complex. Can you imagine the system maintenance required to keep it working long term?

It's not going to last forever and the resources to keep it "alive" are going to be greater and more difficult to coordinate than those required by much of our existing technologies imo.