Deliberate mistake: When the Doctor begins to "fade" in the transporter room his mobile emitter fades with him. Since it's made of solid matter and is not a hologram, this shouldn't be possible.
Greatfellas - S2-E10
Deliberate mistake: When Quinn comes back to the hotel room with a large stack of money he'd won at the casino, he waves it, and you can see a 1-dollar bill in the middle. He was supposed to have over $100,000, but since the crew couldn't get that much money for a brief scene, they used a stack of dollar bills. (00:35:40)
The Ghost Next Door (1) - S4-E3
Deliberate mistake: Right before rising to check out who was calling her name, Hannah sets her laptop, which she was using to write a letter, on the bedspread. The computer screen is shown to be blank, meaning, 1. she switched it off, which she didn't, or 2. it was out of order, which doesn't make sense since she was using it seconds before. (00:02:10)
Deliberate mistake: Because the series changed networks after the end of season one, this episode completely ignores that season's cliffhanger about Harm being arrested as a suspect in the murder of Lt. Schonke.