"I have a white doctor's coat" Like Melissa said, this sounds like you're saying the doctor is white. So it means, I have a caucasian doctor's coat. "I have a doctor's white coat" This sounds like it isn't your coat. You have a coat that belongs to a doctor, but that doctor isn't you. "I have a white doctor coat" This is the best option. "Goodbye to the Doctor's White Coat" is the better headline since "Goodbye to the White Doctor's Coat" sounds like you're talking about a white doctor. Another option is to call it the "Doctor's lab coat". They both work though.

I have a white, doctor's coat and good bye to the doctor's white coat. Good bye to the white doctor's coat suggestions you are calling the doctor white.