Are the advantages of a coaxial rotor on a helicopter also applicable to a ducted fan?
Part of the gain of coaxial rotors on a helicopter is not having to power the tail rotor (IIRC it takes ~10%). So for contra-rotating propellers and fans, which don't need a tail rotor, the advantage is smaller, about 6–16% [1].

Now ducted fans (assuming as used in turbofan engines) usually contain stators. These do not stop the rotation of the flow (that takes away the energy) completely, but they do reduce it significantly and with much lower complexity. The remaining possible gain would probably be at most couple of percent and that is likely not worth the increased complexity and weight of additional gears. Don't forget that turbofans run at significantly higher rotation speeds than propellers.