First time hearing of FitFu but I'm glad to see that the company shutting doesn't doesn't completely make the app useless. I hope more apps think about this as they launch.
FitFu allows you to do a lot of stuff without even signing up for a "FitFu online" account, and even once you have signed up we built it to deal well with being offline or on a flaky internet connection. Effectively when we "turn off" the servers the app should just treat it as if it is offline and continue to function normally but without the network features working. I think this is generally a good approach to take on mobile.