No, you don´t.
With "do" you can ask questions that elicit asnwers with do, or the verb that is in the question.
Example: Do you like ice-cream?
Yes, I do.
Yes, I like.
That is an Achilles´hell to the student that speaks Portuguese, because in questions with "have" he forgets and tries to answer with "I do".
Back to the crux of the question, with "would you rather" is a question that states two choices, one is the one that you is expected to pick.
It´s a choice between two or more things (yes, involves choices, not just that you "like" something. Here the person have to make a choice between them.)
In this way, "do you" don´t go with the adverb "rather". "Would rather'' is the proper way, unless I am proven otherwise.http://www.thefreedictionary.com/ratherhttp://www.wordreference.com/pten/preferir