Good question. AFAIK the onus is on you to prove that you are well/or alternately too sick to fly for insurance purposes. Airlines will get doctors to make the judgement call. Many airports have doctors who can assess you and determine your fitness to fly.

If you were too unwell to fly, I would assume that you would have to get certificates to that effect, pay any rescheduling fees required by the airline and then claim it back on insurance. I don't know for sure though, as I've never had to do it, touch wood. I did once cancel flights due to a family emergency, it was non-refundable but they offered rescheduling. I don't know how, but they didn't cancel it correctly and the flights were adjusted and taken.

Once I did have the flu and developed conjunctivitis during the flight. I used the self-check-in and didn't see any staff. It was a non-refundable trip that I had to take, and I do wish that I hadn't gone if I'd known how sick I would feel on the plane.

I wonder if other sick people do silly things like I did which leads to more emergencies on board (SIN-SYD last night someone required medical assistance, but thankfully not serious enough to turn back to SIN, and I've previously been on a YVR-FRA flight that turned back).