At the talkback, Lauren Mitchell from the original cast said that she thought the show was influenced by being developed during the AIDS crisis, which is not the same as saying it's an intentional direct allegory. Lapine didn't comment one way or the other after her comment.
The second act of Into the Woods has an ongoing sense of death randomly striking people who are just trying to go about their lives. Would it have been written like that if AIDS had never existed? It's an unanswerable question, of course.