Visible crew/equipment: When Professor Higgins is having Eliza practice saying the letter H by blowing into a tube to make the flame waver, the paper she's holding is actually Rex Harrison's lines. Once the paper starts to burn and the camera angle changes, you can read them and it's practically all of his lines for the whole scene.
Visible crew/equipment: During the pre-bomb drop check near the end, there is an exterior shot of the plane. Look in the top-right quadrant of the screen, and you'll see the camera crew reflected in the window glass of the camera plane.
Visible crew/equipment: While Shannon is driving the tour bus, there's a long shot of the bus speeding down the hill, and in the next exterior closeup facing the windshield, we can see the reflection of the camera. (00:26:40)
Visible crew/equipment: The shadow of a camera and cameraman are revealed on the left side while the President (Frederic March) is moving from one room to another (door jamb), after learning about Martin Balsam's death. (01:20:00 - 01:21:00)