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: When Sixsix impacts with the ground due to Greymatter sabotaging his jet pack, the furrow he ploughs into the ground is already present when he skips across the ground.
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: 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."
Deliberate mistake: When one of the yellow fire trucks moves away from the burning Thompson Tower, the back of the truck is set alight by the flames.
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: The Seaview's 8 huge picture windows were a concession to design over practicality. Though they knew that such big windows could never withstand deep sea pressures, the producers wanted the characters - and the audience - to be able to see all the underwater wonders "up close." The "impossible" windows became one of the show's most popular features. (00:25:10)
Deliberate mistake: "The Creature's Revenge" (Adventure 3, Part 5): The 2-inch heels on Sapphire's bright blue boots disappear and reappear repeatedly. They're most often missing in the scenes she appears in with Steel, presumably so that her height won't exceed his. (00:15:15)
Deliberate mistake: Here, as in many other episodes, Doug and Tony are time-shifted wearing period (in this case, Viking) clothing. But they tumble into their next adventure magically wearing their left-behind 20th Century clothes again. (00:45:30)
Suggested correction: A comment, not a correction to your astute and correct observation about the Time Tunnel. Aside from it being episodic television that allows characters to look the same in all situations, I believe Tony and Doug keep their clothing, hair length, lack of facial hair, etc, because the Time Tunnel "remembered" the exact disposition of Tony and Doug when they first entered the Tunnel, allowing them to keep their clothing, hair length, etc., as they slip through time. Any "Time Tunnel" aficionados wishing to add more to this would make a fascinating discussion.
A valid speculation with which I agree, but as the poster himself points out, not a correction.
Deliberate mistake: For reasons unknown and unexplainable, the otherwise spacious and well-equipped Moonbase has halls and doorways too short for the average human to fit through without ducking. (00:20:00)
Deliberate mistake: "Treasure of the Ages": Darien delivers Mary's child during the hurricane. There's no time lapse in the scene, but somehow the kid emerges perfectly clean: no blood, no placenta - and no umbilical cord. Darien also neglects to wash his hands before reaching in to turn the breeched baby around. (Ewwwwww!) (00:45:00)
Deliberate mistake: On several occasions throughout both series/seasons, whenever a Terminator fires a weapon, they blink. This should not happen because that would be a human reaction - Terminators aren't human. They're made to look and act human, but SkyNet don't focus on the little things; such as emotions, breathing and reflexes. The three most notable instances of this are in Season 1, Episode 5: "Queen's Gambit", Season 2, Episode 21: "Adam Raised a Cain" and Season 2, Episode 22: "Born to Run". In "Queen's Gambit", Cameron blinks as she fires a machine gun at the T-888 in the back of the black van as he runs towards her, in "Adam Raised a Cain" the water delivery Terminator blinks as he shoots Savannah Weaver's nanny Debbie and as he shoots the security guard and Catherine Weaver in "Born to Run".
Deliberate mistake: When Wonder Woman is bouncing around in the padded room, there's one part where you can tell the stunt double is not Wonder Woman.
Deliberate mistake: As the Wraith "stun bomb" activates at the SGC and everyone falls to the ground, it is visible how everyone uses their arms to dampen the fall.
Deliberate mistake: Towards the end, Logan is thinking about killing himself and the water drips from the ceiling. He is on the top floor, so nobody would be living above him.
Crisis on Earth-X, Part 2 - S6-E8
Deliberate mistake: During the big battle, Arrow is knocked down and when he opens his eyes, the reflection in his eyes is tilted over to give the sense he's seeing things sideways. But the reflection should still have shown the proper orientation no matter what orientation his eye was at. Think of looking into a mirror and then rotating the mirror to the side, the reflection wouldn't rotate.
Deliberate mistake: In the scene when the timer exceeds 38 minutes, said timer is displayed in big digits on a computer screen. There is a shot of General Hammond with the Stargate in the background. There is a pane of glass between the General and the Stargate. In the shot, General Hammond turns and looks at the gate, and then the camera's focus changes from the General to the Stargate itself. In doing so, a reflection of the timer in the pane of glass is also brought into focus. However the timer is displayed correctly when it should be flipped left to right as all text is when it is reflected in either a pane of glass or a mirror. On the DVD, the writers revealed that this was done on purpose so that the viewer would have a concept of the time running out even when the timer wasn't in shot. (00:17:30)
Deliberate mistake: When the T-Ship is hit from the blast from the exploding space station, it spins around for a while before it re-stabilizes. Just before it does so, look out of Raven's canopy: the background outside with the stars and even a planet spin as the ship does, whereas realistically it should remain calm and stable.
Deliberate mistake: Season 2, episode 6 "Identity Crisis": When Bizarro starts decaying into the parody form of Superman, his costume goes with it; the blue fades, and even the 'S' insignia changes. Strange that superhero garments change right alongside their wearers when they suffer from an unstable genetic effect. Were the clothes cloned right onto his body, instead of sewn?
Deliberate mistake: Several shots have been taken from Series 1, including the group of guards firing in the hangar bay, and various shots of the shuttles during the chase.