Using 無理 to empathise `` seems to be a part of some standard phrases whenever I would wish to empathise with a person who in unwell or in distress. Could someone help with examples of such phrases, and the context where it is appropriate to use.

I'm guessing the phrase you're referring to is and its variants. For example, you'd say this when you make a request of someone and realize that it might be a large undertaking or inconvenience, and you want to express that the person you are asking doesn't need to go to such troubles for your sake. More generally, you would use this phrase when you express hope that someone should not overexert themselves, like when they are sick or under some sort of physical or mental distress.