Similar question have been asked throughout this forum.

The answers have been consistent:

As a rule Lufthansa - as most European carriers - will stick to the fare rules without exceptions made. So if the ticket is "non-refundable" it is "non-refundable".
I personally have never seen any fare rule that would allow for a refund in case of illness - that is what travel insurance is for.

If you need any specific advice on our fare it might be helpful if you share the fare code with us or at least the fare booking class.