"Won't that lead to disastrous consequences for affiliates who depend on cookies to track sales?"
No. Third-party cookies aren't needed to track affiliate sales. The cookie is set and read by the merchant's site when the visitor reaches it.
Third-party cookies are set and read, for example, by AdSense ads embedded in your site. With third-party cookies disabled, AdSense, and other ads served from a third-party, will have a harder time providing ads tailored to the visitor.