I assume that there are philosophical questions or problems that were once hotly
Interesting question. Here is a possible example: In Plato's Protagoras, Plato has Socrates and Protagoras argue about whether there is just a single virtue (despite the different names for virtues, such as "justice" and "courage" and "wisdom" and so on), or whether there are several different virtues, corresponding with the different names. So one might think that Plato decided to take up this topic, because he thought it was one that was controversial or under discussion at that time. To the best of my knowledge, however, no one believes in "the unity of the virtues" anymore. But even if there is someone out there who does, I think it might still be true to say that the view is not taken seriously by "most contemporary professional philosophers," as you put it. I'm sure there are other instances of "dead" theories, as well.