At about 33 minutes in, when Jim is handed $1,000, the face side of the money is shown toward the camera (down from the desk's angle); when he holds it up toward himself, the backside of the money is showing toward the camera.
Even after Jim Phelps cuts the steam pipe to rig it, no condensate drains out. The pipe should have at least had residual fluid in it if it was carrying steam.
Even with makeup and prosthetics, Greg Morris (Barney Collier) clearly looks nothing like the boxer who he is supposed to be imitating. That no one recognizes this is completely improbable as there are photos available of the fighter as is shown during the 'mission lineup" at the beginning of the episode.