SAR teams are generally self motivated, very experienced, highly trained, and go to great lengths to operate safely. As hikermor notes, studies have shown that the most common way that SAR team members are killed or injured is driving to and from the incident, and in aircraft accidents.

As Teslinhiker notes in another thread, SAR missions are commonly suspended when conditions become unreasonably dangerous for rescuers. This depends a lot on the circumstances. Sometimes dangerous missions will be done when there is a high chance of saving a life. For example when it is known that the person is still alive. When it is known or strongly suspected that it is a body recovery mission, SAR teams tend to get much more conservative. A classic "Risk vs Reward" situation.

One situation that sometimes vexes SAR teams is when non-trained "spontaneous volunteers" show up and want to help. These are often friends and family of the subject. Another case when lot of people often show up is when kids are involved. While the extra manpower can sometimes be helpful, the problem is that these spontaneous volunteers are often themselves inexperienced and poorly equipped. It can be very hard in a fast developing incident to sort out who can be an asset to the mission, and who is more likely to add to the problem.