This should help. From another thread:
This is howyou can see your podcast “playlist” queue: start in the Podcasts App then:
1. In ’Listen Now’ play your first episode.
2. at the bottom of the screen tap on the currently playing episode to bring the episode to full screen format
3. Swipe up starting your swipe about 1 inch from bottom of screen
4. If you have made a “playlist” you will see the episodes in your queue under the heading ‘Up Next’
5. you can reorder the queue by touching and holding the 3 horizontal lines of an episode in the queue and sliding it into your desired location.
It should play continuously....