Should a citation be included when answering a question regarding a specific source? For example, "How do the six types of shenanigans relate to the ethical use of a flux capacitor according to Smith and Smith?"  (Smith and Smith are authors of the textbook).   For this type of question, where there is a specified source that must be used, is it necessary to re-state the source in the answer?

The question is posted in a discussion board.  Only students of the course and the instructor are the intended audience.
Also, the chapter of the book is directly stated in the subject line of the posted question.
Ex. "Chapter 2: Electric Boogaloo"
If it is a discussion board the audience are only the course instructor and the students of the course, whether or not you repeat the citation in your answer is not going to have any important or lasting effect. That said, while a very strict instructor might complain if you do not repeat the citation, I cannot imagine any way in which repeating the citation could have a negative effect. If you are in doubt, just cite it again.