Near the end of the 2nd underwater scene in the salt water pond where he's trying to detonate a bomb, Jim West comes up out of the water with his hair wet and matted down. Seconds after the bomb detonates and the camera is back on him, his hair is not only dry, it's completely blown dry.
The sheet of paper that Artemus throws darts at is creased and wrinkled. Then, in a later shot, that same sheet is crease- and wrinkle-free.
At 41:50, the metal grate breaks in different spots than where the corrosive fuse was and before it fully burned through all the grates.
Starting at 22:20, James is drawing a secret message to Arte on a menu, but the menu is taken by one of Elisha's men. Arte then lifts the tablecloth to reveal a carbon paper copy. However, a clear carbon paper copy couldn't have been able to be made with a pencil through a heavy paper menu and 2 layers of tablecloth.
After Artie cuts the metal door, when it swings back, the cut is too smooth to have been made by the torch. If the door had indeed been melted by the torch, the cut would have had blobs of melted metal around the cut on both sides of the door. In addition, the width of the cut is far too thin to have been made by the torch - if it had been made by the torch, the metal would simply have re-sealed itself as it cooled.
The flame from the torch Artemus Gordon uses to get through the steel door is too diffuse to cut through metal. Cutting torches need a narrow, highly focused flame.
James West is diving in an inland saltwater lake, yet a sawfish swims past him. Sawfish are only found in the ocean.
In the second underwater sequence (the upper tank) where West is retrieving the package, the same sequence is used each time West dives underwater - the same fish swims by at the bottom of the screen at the same place, West kicks and swims exactly the same way, etc.
While disguised as a postal worker, Artemus invokes the familiar "Neither snow nor sleet" motto. Though this saying can be traced back to the 5th century BC Greek historian Herodotus, it didn't become associated with the US Postal Service until 1914, when the architectural firm of McKim, Mead and White inscribed it on the facade of the New York City Post Office.
The darts Artemus throws at the paper have brass barrels and plastic shafts and flights. These did not exist until well into the 20th Century.
Crystal uses the term "superman." The term "Superman" was coined by George Bernard Shaw in 1903 to translate "Übermensch," a German term, though the noun superman was used as early as the 1890s. The term wound up a fixture in the common lexicon after Thomas Common's 1909 translation of Nietzsche's material.
Near the end of the underwater scene in the salt water pond when Jim West is facing the camera, a man in full scuba gear is visible behind him (directly over his left shoulder).
There is no way they could have sealed the gas in the cave. It was already escaping with them standing there. If the gas was that deadly without a Hazmat suit they could not have gotten close enough to seal it, with the gas already in the entrance.