If the faucet is already running, the most suited to your situation is the first: Why did you turn on the faucet?
Why are you turning on the faucet? Would be used if your friend is about to, or has just begun to turn on the faucet.
Why do you turn on the faucet? Would be a very rare, usually rhetorical question that I would only expect to hear from a parent talking to a child or a teacher talking to a student.