First class functions pretty much have no bearing on whether or not interfaces are useful. Sure, one use case for inner classes is to implement interfaces in-place in situations where what you really want to do is just pass a function, and in those cases it's probably better to use a function instead, but the equivalence doesn't go the other way.

There are plenty of cases where you really need to pass an object to a function and know that the object supports multiple operations, for instance take a look at Map<K,V>: http://docs.oracle.com/javase/6/docs/api/java/util/Map.html

There are 14 methods there, and when I write code that takes a Map object, I really mean it - I'm not just using the interface as a hack because I want a function pointer, I need an object that supports all of those methods, and I'm probably going to be using several of them.