We are hosting an SaaS website and API with the https protocol, both of which are accessed by numerous users with a lot of different browsers and client-side apps. What would happen if I made PFS mandatory in my IIS Web Server's configuration? Could there be problems with incompatible clients which do not know how to handle PFS?
All major browsers support ECDHE: IE since IE7 (on Vista and newer), and Chrome and Firefox for even longer. However, Windows XP users of IE do not get ECDHE support, and even though Windows XP is past end-of-life, people are still using it. If you want to allow XP users, then I suggest just putting the ECDHE modes at the top of your cipher preference string. If you value PFS over Windows XP users, then create a cipher preference string with only ECDHE modes.
Qualsys's SSL Labs has a great article on deploying PFS, and of course their checker will help you figure out which browsers will negotiate with which cipher suite.