What do I do if after I answer the bona fide residence dates, it continues to request this even after I answer it?

In several sections of TurboTax, the questions will reset each time you re-enter the interview section.  

To verify that the answer is "sticking" after you finish the Foreign Earned Income and Exclusion section click on Review on the left side of the screen.  If the information is missing still, TurboTax will alert you with an error.