Yes, chimney fires can make a metal chimney unsafe. High temperatures can heat the pipe until it glows red or worse. At this point, oxidation can start. Creosote is acidic, so oxidized metal can be further corroded. Different qualities of stainless have different properties, so there is no hard answer of yes or no. Stick a webcam on a stick and take a look inside, and if possible between the two pipes. If you see burn-through you have an answer. If you see the outer pipe metal is discolored, you also may have a problem, depending on how it was installed.
When I installed my metalbestos chimney in Vermont, I made sure that ALL adjacent parts were metal (thimble and where it went through the roof) and that I could literally hold my hand on any nearby wood when the stove was going full blast. My requirements were far more than the 2" offset required.
My current metal chimney is not as well installed, so I use a brush to make sure I have no chimney fires.