Plot hole: In the heat of killing fake Red John, Jane forgot to notice that Craig O'Laughlin never knew about the mall setup to lure Red John. So definitely Gale Bertram himself was in contact with the serial killer which he may have missed first up, but all that time in jail afterwards he could have figured it out.
Character mistake: When Kim is trying to convince Lisbon to help the FBI on their case, she says Jane stopped the ten-year reign of a mass murderer. Red John was a serial killer, which, by the FBI's definition, is a criminal behavior distinctly different from mass murder; he may have had a large number of victims, but he killed them either one or a couple at a time, with a cooling-off period in between, not all at once.
Deliberate mistake: When the investigators are in the basement wine cellar, they turn on their flashlights to illuminate the pitch black area as they walk around. However there is a light source moving with the camera shining on them as they are casting a shadow behind them.
Other mistake: When Jane is setting up for his escape from prison, he captures a wild mouse. Then he presumably holds her in his hand for the whole time. I've handled mice, and when they're scared, they bite hard enough to take chunks out. There's no way a wild mouse would stay in human's fist that long without drawing blood. (00:31:50 - 00:33:10)
Continuity mistake: When Jane speaks to the owners of the restaurant, the female owner is writing something and also has a drink. She takes the drink with the pen in the same hand. When the shots changes, the pen is no longer in her hand. The shot goes back to the original position and the pen appears again. (00:09:35)
Factual error: The container for the dangerous biological material is marked "Biohazard Level 5." There is no level 5. According to CDC regulations Biohazardous materials are rated 1 through 4. Based on the description of the material in the episode, it would be required to be stored in a laboratory rated at Biosafety Level (BSL) 4. All BSL 4 labs require personnel to wear a fully enclosed, pressurized suit when inside the lab. (00:01:05)