In the scene where Lou Diamond Phillips is being dragged under the car, Philips actually did the stunt himself. He stated in an interview he had marks on his back for months after filming was complete.
In several scenes Max suffers from heart problems. In a later interview, Fred Gwynne said that he actually had a real heart attack while filming one of those scenes. The cameras kept rolling as they called 911, and the director decided to keep the footage of the real heart attack in the movie. Fred got out of the hospital a few days later, and managed to finish the filming on schedule.