The first time Briggs' name is mentioned on the recorded instructions, he is called Mr. Briggs. The second time, he is called Mr. Reese.
Dan Briggs puts his suit jacket on over a shirt with sleeves rolled up to his elbows. A few seconds later, when he's putting on his tie, his shirt sleeves are shown under his jacket sleeves, unrolled and with the cuffs buttoned.
The voice on the record says the record will decompose one minute after the seal is broken. It's actually closer to 90 seconds.
When Dominguez slams the door on Terry Targo's hands, freeze-frame makes it clear that actor Wally Cox is holding a pair of wax hands by the wrists and has inserted the fake fingers into the door.
The rubber Briggs mask worn by Hand has lips that are fused shut. These would have made it impossible for Hand to talk while impersonating Briggs.
On Roland Hand's promotional photo, shown when Briggs is selecting his team, the word Disguise is misspelled DisQuise. Unlikely that a world-famous performer would have such a glaring mistake in his advertising.