Not being immortal (in the sense of dying from old age) is obviously an illness, but hasn't been recognized as such by most outside the transhumanist community, because it's universal. It would be in a sane society, but there you go.

Is the set of all possible fatal illnesses itself an illness? I don't think so; that just seems like a type error. Lack of immortality is a bad thing that society ought to take steps towards fixing, but calling it an illness is just mixing up terminology.