Deliberate mistake: The show regularly shows characters by the Chicago River and either on or near Michigan Avenue, when they are getting lunch or walking to work. Both U of C hospital (where the hospital exteriors are filmed) and Cook Country hospital (where the characters are supposed to work) are about 5 miles from this point. This would be an extremely long distance for them to walk on a routine basis. I'm sure this is done just because Michigan Ave and the river are two of the most scenic parts of Chicago.
Battlefield - S26-E1
Deliberate mistake: Look for the "creases" in the wall inside the TARDIS, which look as if the wall was just a sheet hung up behind the console. (In fact it actually WAS a sheet and not a "solid" wall - the TARDIS wall prop was apparently unusable due to water damage, due to poor storage between seasons. This also explains the lack of TARDIS interior scenes throughout the rest of season 26: it was not thought worth repairing the TARDIS set [or building a new one] when the show was in imminent danger of being cancelled after season 26).
Deliberate mistake: These errors I assume were caused by the show running 3 times longer than the real war. Col. Potter arrived at MASH in September 1952 a year before the war ended, then a few months passed by Margaret got engaged for 8 months, then Frank Burns left and Charles took his place. In this time the war would have ended. In the final episode Margaret mentions Charles has been there for 2 years. Which means he would have arrived in 1951 a year before Trapper and Henry left and Col. Potter arrived. It's a lot to take in.
Deliberate mistake: Because Paul Darrow had turned an ankle while shooting the fight scene with Gunsar, the uninjured Avon is inexplicably limping throughout most of the fight. (00:14:00)
Deliberate mistake: This show is famous / infamous for having many long conversations between characters that are walking the corridors of the West Wing of the White House. In this episode, the chat was long enough that more space was needed to complete the walk, so the actors were instantaneously teleported to a different spot to lengthen the walk. Using http://www.whitehousemuseum.org/special/wwtv.htm as a guide, Pres. Bartlet and Leo leave the Oval Office, pass through the Presidential Secretary's office, turn left, passing between the Roosevelt Room and the Chief of Staff's office. At the end of the Roosevelt room, they turn left through dark wooden doors that are open. Instantly, the actors are at the next junction higher on the map (no wood doors), as if they'd just passed between the Roosevelt room and the Communications Bullpen, and they continue (downward on the map) through the intersection where they original turned (watch for the doors after Leo says "You wouldn't understand"). (00:02:40)
Deliberate mistake: The credits list "Number 86" as played by John Maxim. But no such character appears in the episode. Apparently, 86 was left on the cutting room floor, but the titles weren't altered accordingly.
Deliberate mistake: In one of the later episodes of season 2 Turk is seen being bored in surgery and checking the patient's wristwatch to see how long they have been operating. Patients are not allowed to wear jewelry or watches during surgery as there is a VERY high risk of infection and it might kill them.
Deliberate mistake: After Ramsey shows Sansa her flayed northern friend, when it cuts to Davos riding the horse through the encampment one of the shots is flipped. Note his sword and his cloak's crossed straps have switched to opposite sides.
Deliberate mistake: When Mary gets to the school where she is going to be teaching, all the children are standing outside wearing shoes. However when they come in to the school they are barefoot. I have a feeling this was done to show how poor the community she was teaching in was supposed to be, but it looked like too cold and rainy of a day to let the actors be outside barefoot.
Deliberate mistake: The newspaper that shows Petrelli's victory has repeated paragraphs through the article. The article seems to be composed of three paragraphs. One paragraph starts with "I was," and two paragraphs start with "The."
Cold Stones - S6-E11
Deliberate mistake: Tony smashes AJ's front windshield of his Nissan SUV one time with a plastic sports helmet and it leaves a hole 4 times the size. Not only is the scale size of helmet to hole incorrect, but front vehicle windshields don't shatter like that. Modern windshields are made of laminated safety glass, which consists of two curved sheets of glass with a plastic layer laminated between them for safety, and bonded into the window frame. It would've taken many impacts to cause such a hole, not one hit. (00:49:00 - 00:49:45)
Deliberate mistake: In the flashback to Angelus rising from his grave, his and Darla's breath is visible as they exhale. This was going to be removed in post-production, but left in due to budget restrictions.
Mr. Garrity and the Graves - S5-E32
Deliberate mistake: Garrity blackmails the town citizens into paying him not to resurrect their unlamented "loved ones." Amazingly, every man in the saloon just happens to have hundreds of dollars ($500 - $1200 each) in his pocket to pay up with. In 1890 money, that's roughly equivalent to a group of modern-day bar patrons all carrying ten to thirty thousand dollars around in their pockets. The scenario was apparently used in spite of its incredibility just to more quickly advance the plot.
Deliberate mistake: After the party, in the scene where Lucas goes into the cafe, he finds Haley crying. The shot is a distant shot with Lucas and Haley in it. Haley's hands are covering her face. In the next shot, just of Haley, her one hand is over her forehead.
Yankee White - S1-E1
Deliberate mistake: Fornell doesn't recognize Gibbs, but subsequent episodes establish they have a long history together.
Episode 306 - S3-E6
Deliberate mistake: In the locker scene where Brian confronts Ben about his steroid use, there are no obvious objects on the lockers (such as slats) that would make the cut and bruise on Brian's back. Brian is not tall enough to have been cut by the top of the locker and Ben didn't pick up in his tirade. (00:25:25)
Deliberate mistake: The main character is called Anthony Dorian in the pilot. But without explanation, his name becomes Anthony Blake in the first regular episode. (00:05:15)
Second City - S1-E12
Deliberate mistake: Kermani finds a nitroglycerin tablet in the suspect's apartment. Because we need to see it when he holds it for the close-up, the tablet he has is much too large to be nitro. That drug is dispensed as an extremely tiny pill designed to dissolve instantly under the tongue. (00:22:30)
Deliberate mistake: Elsa's dress in Season Four is different from the dress she was wearing in the Season Three finale. The new dress is a different shade of blue, and has a completely different design, even though the present day events of Season Four take place mere moments after the end of Season Three.
Deliberate mistake: While Megan O'Hara is undergoing breast reconstruction surgery, as the camera pans a close-up side angle view of her face you can see she is wearing make-up, specifically eye make-up, which is strictly prohibited in a surgical sterile environment in the United States, especially in pre-scheduled surgeries. (00:23:50)