At the beginning the heaters change their positions. When Mrs. Pennell pulls the heaters out the one closest to the bunk is pointed to the viewer's right and the other one to the viewer's left. They remain this way until she goes to bed and Mr. Pennell begins dictating his report into the tape recorder. Then when they show the beakers of plankton starting to bubble over, the two heaters are pointing in opposite directions_
In the laboratory aboard the Seaview, Adm Nelson is using a Zeiss "Jug-handle" microscope. There were made between 1898 and 1914, so it was already an antique during filming in the 1960s, let alone during the future dates the series is set.
At the beginning Mrs. Pennell says they are near the South Pole on a ice floe. This is only possible at the North Pole where it is all water instead of at the other pole, in the middle of the continent of Antartica. This is further verified by the fact the pilot who contacts Seaview tells them he's from Baffin, which is located in the far north of Canada.
When Mr. Pennell attempts to contact the Seaview he never presses a button when he is talking. In standard transceivers if the person doesn't press a button the equipment doesn't "know" to transmit what is being said, it will simply stay in reception mode.
The pilot uses an incorrect call sign when he hails the Seaview. When he gives his call sign he says, "J for Jenny, K for Kilo, S for sugar, one-four-nine." This should have been an immediate red flag to Capt. Crane, as no reputable pilot would have used nonstandard words. A real pilot would have simply said, "Juliet Kilo Sierra one four niner", using the correct alphanumeric code for letters and numbers.