Do I need to mention having survived a life-threatening illness when applying for a new academic position?
If you think it helps explain a work or productivity gap in your career, I would mention it and I would do so in writing rather than let your prospective employers speculate. Because overcoming a major medical challenge can help you align your priorities and strengthen you in other ways, doing so can definitely be done in a way that leads one to conclude that it is a strength, not a weakness, of your candidacy.

If your previous condition is not relevant in these ways, I don't think you have any obligation to bring it up. Sure, your illness might return but nobody is 100% immune from serious illness impacting their ability to work or be productive. For that matter, severe chronic illnesses that unambiguously affect productivity (e.g., cramps, migraines, etc.) are simply not the kind of thing that people bring up while interviewing and candidates have no responsibility to do so.

Reminding prospective employers of this can open to the door to (illegal) discrimination based on your medical history and I don't think you are helping either yourself or your prospective employers by bringing it up. When you take an offer, it might be good to let your future department know about your history as background but you might even let this just be raised socially.