UPDATE: After making sure it wasn’t a plugin-compatibility issue, I refreshed my permalinks via the instructions here: https://bbpress.org/forums/topic/bbpress-not-working/.
And it worked!
I’m having this exact same problem, too. Only, for me, this issue has appeared out of nowhere on an existing install. I’m running bbpress on another website and compared the backend settings; it’s working on one site, but not the other.