Basically, が is the particle used to identify things. In this sentence, we're talking about Tanaka's book. We need to identify which one is his and we'll mark the location of that book in both the question and the answer using が. More here: http://www.guidetojapanese.org/learn/grammar/particlesintro

Hi, Matt has explain very well and the correct way of asking is: どの本が田中さんの本ですか。