What disease did Dr. Rank inherit from his father in A Doll's House?

Dr. Rank inherited syphilis from his father.

That he inherited this specific disease matters in several ways. First, because Rank is aware of his condition, he cannot follow his affection for Nora (if he was ever going to). This means there is a limit to how close they can be: it increases Nora's isolation. Since syphilis advances through stages, it also means that Rank is removed from the drama and the relationships, making her more isolated.

Second, it fits his name. He is rotting from within, and so he is rank, in addition to being Rank.

Third, the disease shows how male actions can shape (and distort) the lives of men as well as women. It is a partial parallel to Nora living her entire life in a doll house, which is something she got from her father and Torvald. As she is limited by male action, so is Rank.