In my experience, upgrading to newer Tomcat versions has always been pretty painless, especially since Tomcat 5 from when on configurations have stayed pretty much the same. If you're upgrading anyway, you might as well go straight to the latest 8.5.x version. 8.5 needs Java 7, though, whereas 7.0 got by on Java 6. But you should probably be running on Java 8 anyway, so that might not make a difference.
I had some difficulty a while back with an application that I wanted to run on Tomcat 8. The difficulty was an incompatibility with some old Spring 2.5 MVC code that was being used. It took me a few days to update all the Controllers to use the more current annotation driven setup rather than the old extend and override some Spring Controller class. Once done I was able to remove the dependency on the offending incompatible library.
Other than that it the transition from Tomcat 7 to Tomcat 8 was painless.