Deliberate mistake: Early on, The Flash's immense speed caused his clothes, and those of people he was carrying, to catch fire because of the air resistance. Gave him an excuse for the suit, but this effect was fairly quickly forgotten, with him frequently carrying civilians or running out of costume with no ill effects at all.
Suggested correction: This is half-true. Barry's connection to the Speed Force allows for some sort of shield around his body that protects him from harm while running. In the beginning, his bond with the Speed Force is weak, whereas later on, he becomes closer and closer to the Speed Force, allowing him to run faster, defy laws of physics, etc.
Going Rogue - S1-E4
Deliberate mistake: When Capt. Cold is about to shoot Joe in the train station, Eddy "shoots" a pillar next to him to make him run. Eddy then comes from behind the pillar that got shot with no cuts. He would have had to been on the complete opposite side to shoot the pillar where it was hit. (00:32:50)
Suggested correction: I watched the scene several times and nothing indicates the pillar was shot from the left side. It looks like the pillar was shot from behind (which is why there's no hole on the side or front), exactly where Eddie would have been standing.