I recently reset my iPhone and restored it using my latest iCloud backup. I noticed in the podcast app that four Christian podcasts appeared under a "Not Subscribed" section (seemed interesting that all of the podcasts had the same general theme). I had never subscribed to, or even heard of, the four podcasts. I deleted them and closed the podcast app. Now each time I reopen the podcast app, one of the podcasts continues to reappear and attempts to start downloading an episode.
I found this question that sounded oddly similar, but it had no answers.
Update
I tried removing all podcasts and syncing with iTunes on my computer (which shows no current podcast subscriptions), but this random podcast still appears.
Best Answer
I'm not sure why this worked, but I reset my iPhone a second time and restored it from the same iCloud backup and that seems to have fixed it.