Character mistake: As Reddington sets up the blood transfer between himself and Ressler he inserts the intravenous line in Ressler's arm first and then in his own arm. This way the remaining air inside the line gets into the recipient's body and could very well kill him. Judging the expert way he handles the set up he should know to first insert the IV in his own arm, let the line fill with blood and then insert the IV in Ressler's arm.
Suggested correction: That amount of air would probably not cause any problem. It actually takes quite a lot of air to cause any issue.
Character mistake: After Elizabeth and Alexander Kirk crash-land into the Gulf of Mexico (technically, the Straits of Florida), Lizzie gets on the radio to report mayday, along with their coordinates. She incorrectly states their position with the following statement: "Mayday, mayday. Plane down off the coast of Havana. 23 degrees, 52 minutes north, 82 degrees, 49 minutes south. Mayday, mayday, mayday." Those coordinates specify two different latitudes (one above the equator, and one below), but no longitude! Luckily, she DID say they were off the coast of Havana, so the Coast Guard would be able to deduce the correct coordinates of 23' 52" north, 82' 49" west (not south!). (00:05:45)
Character mistake: After the seaplane crash lands off the coast of Havana, Elizabeth radios a mayday. However, she broadcasts nonsensical coordinates. The first set she provides is 23°52' north, the second set 82°49' south. These combined coordinates can't possibly plot to any location on earth. The more probable coordinates should've been 23°52' north, 82°49' west. These coordinates would put the plane roughly 57 miles NNW of Havana, Cuba. (00:05:48)
Character mistake: Tom's toe tag says that he was 33 at the time of death. In the next episode, Ruin, his Tombstone says he lived from 1985 to 2017, making him 32 at the time of death. (00:41:55)
Abraham Stern (No. 100) - S5-E11
Character mistake: Reddington says that the 1943 pennies he is hunting are special because they were made of bronze when all other pennies were made of copper. The 1943 pennies were actually the steel pennies from WWII as to save copper for the war efforts, there were some copper errors that year, but the majority were zinc and not copper.
Pattie Sue Edwards (No. 68) - S5-E15
Character mistake: When Aram wears the CDC hazmat suit, the CDC personnel advises Aram that he has 20 minutes of air to keep him alive, making the mission time critical. What she says additionally is that "oxygen levels plummet and carbon monoxide (CO) levels rise" CO is not a byproduct of human respiration - carbon dioxide is.