As a man is supposed to respect his wife more than he does himself (I presume that includes rabbonim also), it would make sense that they can and should do such mundane tasks as take out the garbage. I had this discussion once with my own Rov many years ago (from an entirely different discussion point), and he told me he takes out the garbage. I said, “REALLY, Rebbie!!??? But you are the Rov of the shul!” To which he replied, “Yes, but in my house I am the husband of my Rebbetzin.”
My basic rule is, whoever is able and available to do it at the moment when it needs to be done, should do it. No one should stand on ceremony. we all make the garbage, so we are all responsible for getting rid of it.