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)