Jim Phelps listens to the mission message only a few feet from where there is a great deal of tourist traffic passing by. There are also people within full view while he plays the tape recording without any type of ear phones to block any random citizen from hearing it being played.
When Barney makes contact in front of the dress maker, a blue Mercedes pulls up and lights and crew are visible in the reflection on the side of the car.
When Jim gets out of the car in the warehouse, the boom mic is clearly visible for a second.