Although as many say its not mandatory to mark the fields as final I would say atleast in one case that I can think of you need to mark the fields as final and that is in the case if you want to make your immutable class thread safe.
According to Java memory model:-
An object is considered to be completely initialized when its constructor finishes. A thread that can only see a reference to an object after that object has been completely initialized is guaranteed to see the correctly initialized values for that object's final fields.
JLS final thread safety
So if you have an immutable class with instance variables which you initialize in constructor as non final variables then there is no guarantee of thread safety as the writes to the non final variables may not be visible to other threads even though the constructor has fully run ( Note recreating this is very difficult as this may occur in a highly concurrent application)