this is an Interesting question, because both quantities depend on integrals over the interaction potentials. Chapmann and Cowilng book discuss a little this aspect.
The binary diffusion coefficient is the reverse of the first order collision integral (see the attached paper for explicit expression).
On the other hand, the excess energy depend on the virial coefficients that are integrals of the interaction potential too.
However the relation, if exists, should be very complex, because the dependence of argument of the integrals in the two cases are different.
Things become even more complex if you consider higher order diffusion coefficients and virial corrections.