Continuity mistake: Phelps takes the tape from the recorder's right-hand spindle. When he opens the acid vat, we can see the tape he just removed back on the recorder - but on the left spindle. The tape he then drops into the acid doesn't remotely resemble the one he pulled, and when he drops it, the recorder's left spindle, where the full tape was in the last shot and where the empty reel should be, now has nothing on it at all.
Continuity mistake: While Barbara Bain is opening the puzzle box, Martin Landau is removing pieces of his disguise. First, he has a handkerchief in his right hand. Next, the handkerchief is gone and he is peeling off his left leather pocket, and the left leather lapel is already gone. Next, he removes his mustache with a handkerchief. Then, while removing the left leather lapel, the left pocket is back in place.
Continuity mistake: When the Major checks the prison log book, the box next to 27 where the time entry should be is blank. He closes the book and holds it under his arm while conversing with a subordinate, and when he opens it again to show it to the other man, the time box has, impossibly, been filled in.