What should I do if my advisor urges me to do a project that I have no interest or faith in?
Sometimes it is necessary to "bite the bullet" and demonstrate that the method doesn't work. But you need to provide clear and irrefutable evidence in a way that can actually convince people, not just a vague claim.

A classic example of the failure to do this is the Challenger explosion: all of the information needed to delay the launch was known, but because the engineers who had the data didn't present it in a clear manner–just a raw output of data—their supervisors overruled them. If they had come in with plots showing failure of the O-rings as a function of launch conditions, anyone could have recognized the danger of the situation.

So if your advisor is insisting, you probably need to demonstrate why his approach  is infeasible or he will continue to insist upon it.