Yes, the methods used on the Biggest Loser are healthy, because the people on the show are taught about having healthy lifestyles. It only seems unhealthy because it is in a controlled environment, so there is a lot of progress very fast. The people have nothing else to do while they are on the show.
The methods on "The Biggest Loser" are healthy because the people are closely monitored by trained professionals with years of experience. Being a personal trainer takes more than just going to the gym a lot. Knowledge about physiology, kineseology, and many other bio-sciences are necessary to fuel the body, and keep it maintained during heavy weight loss. Without the proper management that is often mostly off-camera, the methods are very dangerous.
Losing that much weight that quickly is not healthy. A lot of those people return with destroyed knees and little knowledge of a sustainable life style. It perpetuates the idea that weight loss is a short term goal. A healthy weight comes from a healthy life style, not from working the body to the point of vomiting.
The short amount of time that these contestants are on that show to drop over a 100 lbs in a matter of weeks and months. It is proven that the quicker you loose weight that the quicker your body is to regain the weight, however if you take years to loose weight and change your mindset, your thoughts, and your choice in foods you can and will be successful in your weight loss program. Drastic weight loss like this doesn't help anyone at all.
I don't think it's healthy, mentally or physically, to go on these reality shows and try to lose huge amounts of weight in the least amount of time. Two pounds a week is the maximum safe weight loss. The contenstants on that show obviously lose a lot more than that. (But do they keep it off, I wonder?)