I don't think there is a way to turn that message off. There is a few things that I am trying out to see if the message won't show up again. 

1. Click Settings…
  - set it to *Check for new episodes every day*.
  - Settings for: *Podcast Defaults*.
  - When New episodes are available: *Download All*.
  - Episodes To Keep: *All Episodes*.
2. Click the *Refresh* button (next to the settings button).
3. Click on ANY podcast, and *select all* — <kbd>⌘ cmd / ctrl</kbd> + <kbd>A</kbd>
4. With all the podcasts selected
  - right click & choose *Update podcast*.
5. *Select all* again
  - and in the same right click menu, select *Do not Auto Delete*.
6. *Select all* again
  - and in the same right click menu, click *Mark As Played/Watched*.
7. *Expand* every podcast
  - make sure you select *Get All*, if they have not been downloaded.
8. *Quit* iTunes, and relaunch it.

Do all of this every day. I am concerned that iTunes won't save the library file and other settings if I don't relaunch iTunes every day.

So pretty much whenever there is a blue dot or whatever indicating you have unwatched/unlistened podcasts, simply select "Mark as played" and hopefully it will keep updating the podcasts by itself. If you are still concerned, move the podcast files in another place… perhaps to an external hard drive. This way if you are too busy to watch podcasts for a while, but still would like to watch them later, you will know for sure that they are safe and iTunes has not eaten them!