The first is that suicide is illegal, so assisted suicide is also illegal. In most, possibly all us states, it is illegal to attempt suidicde. At the very least, the person can be considered "a danger to themselves or others" and forcibly hospitalized. So why would we make suicide legal if the person has a doctor assist them.
Second, and more philosophical, is the belief that the duty of doctors is "first, do no harm." Intentionally taking a life violates the physician's oath.
Finally, there is the slippery slope argument. If doctors are permitted to assist an individual's suicide, what happens when the family of a patient -- a patient who is unable to speak for themself -- asks the doctor to perform a suicide procedure? In the extreme, opponents cite Nazi Germany, where the state deemed handicapped and mentally ill people as subjects for termination.