Currently, if there are no questions with active bounty on a site, the site's homepage and question lists do not have the "featured" tab at all:
However, navigating directly to the URL of the featured tab results in this:
I find it rather helpful and good, and it encourage users to use the bounty system, so my suggestion is to always show the featured tab, even when there are 0 questions with bounties:
Keep in mind, the number of questions with bounty always show in the "featured" tab already:
But in blue, so marking it red when there are no questions should make it clear we should not expect it to list any questions.
This will also be consistent with other elements that are always visible even when they can't be acted upon, e.g. edit link when post can't be edited, vote buttons on own posts, etc.