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.
Dr. Rank, a close and old friend of the family who visits Torvald and Nora every day, has spinal tuberculosis, a disease Nora thinks was caused by his father's lascivious lifestyle, including many mistresses and too much food and drink. Nora refers to the doctor's illness as "spinal consumption" and explains that the doctor has been sickly since he was a child. Dr. Rank is clearly in love with Nora, but their affection goes nowhere. In the end, Dr. Rank dies of his illness at the same time that Nora's marriage to Torvald dissolves. As Nora and Torvald separate, they receive the doctor's card with black crosses in the mail, signifying his imminent death. Dr. Rank's illness is symbolic of the illness between Nora and Torvald, as the doctor has dedicated himself to them and their marriage. When he dies, so does their union.
Although in the play Dr. Rank's ailment is called "spinal consumption", critics generally agree that the disease is actually syphilis. The Doctor inherited the disease from his father, who was known to have lived a dissolute life with a number of different mistresses.