While the Plex app can already download new podcast episodes (with varying amounts success in some cases), there’s no way of knowing when a new episode has been made available without opening up the app. Because this kind of background downloading is already going on, it would seem plausible to, at the same time, notify the user via Push Notification that a new episode is now available (and subsequently downloaded). I’m not sure that the Plex mobile apps have any sort of Push Notifications ability right now which could complicate things, but I think this would be a nice feature that is available on nearly every other podcast platform.
(Credit to Reddit user u/SirSirae for the original suggestion)