This error occurs because the item you are trying to delete or unpublish is "featured". You can tell which items are "featured" by looking for a yellow star next to it.
In order to delete/uppublish this menu item you have to make something else "featured". To do this you will need to click the white star associated with another menu item which will turn it into a yellow star.
Once another item has the yellow star next to it you can click on the yellow star associated with the item you want to delete and it will become a white star.
Noew you can delete/unpublish the menu item.