For taxes AND naturalization and DL and car registration you should claim to be a resident of B. Your car and DL are supposed to be registered where you live (state B)*, not where you work, and not some third state where you neither live nor work. And you'll pay taxes to state C as a nonresident (and perhaps some taxes to B as a resident, if the formula B and C use to offset the double taxes doesn't completely cancel B's taxation).
How does state A come into the picture, if you neither live nor work there? If you get into a car accident, you're going to be in trouble with the law and with insurance once they find out you're not living in A. Are you a college student there?
*if your car is normally not parked in the state where you live, you may be allowed to register it only in the other state where it is regularly parked overnight. For example, snowbirds who live in New York but keep a car in Florida where they visit 4 months per year.