Featured Content: settings being overridden by wp-admin changes

User in the support forums reporting that their settings for Featured Content is being changed whenever they make any changes to their settings in Settings -> Reading in the wp-admin dashboard.

Thread: http://wordpress.org/support/topic/saving-reading-settings-resets-featured-content-setting?replies=3

I have not been able to duplicate her exact issue where the tag field is blanked out, but have had similar results.

Steps to reproduce (using Twenty Fourteen):
1. In Appearance -> Customize, set the featured posts tag to something other than the default "featured" and check the box to hide that tag: http://d.pr/i/JBQ4
2. Go to Settings -> Reading and change one or more things and save. I've tested changing the blog follow email text, changing the feed from Full to Summary, and hiding the "Related" header for related posts.
3. Go back to Customize and notice that your settings have been reverted: http://d.pr/i/SYDH

The choice of layout (Grid or Slider) was not changed in my testing.