At about 33 minutes in, when Jim is handed $1,000, the obverse side of the money is shown toward the camera (down from the desk's angle); when he holds it up toward himself, the reverse 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 he is supposed to be imitating. That no one recognizes the discrepancy is completely improbable as there are photos available of the fighter as shown during the 'mission lineup" at the beginning of the episode.