Abner Come Home / The Sewer King - S1-E16
Audio problem: When Arnold is playing chess with the Sewer King, Arnold jumps across the table and knocks the King over. This sequence appears to be sped up: the toilet crashes way after it is destroyed, and the King says "Noooooooooooo," yet his mouth never moves.
Suggested correction: The Sewer King's mouth does move, but it does say the line a little too late. However, the toilet crashing sound is exactly where it should be, and the scene is definitely not sped up.
Magic Show / 24 Hours to Live - S1-E18
Audio problem: Helga rides past Arnold in a bus and yells "24 hours until you die!" or something like that. Her mouth never moves- it just stays open in an evil smile.
Suggested correction: Although it's hard to make out, her mouth is moving when she says that. Also, she says "23 hours until you die!" not 24.

Tutoring Torvald / Gerald Comes Over - S1-E13
Continuity mistake: Arnold switches from his left hand to his right hand when taking the test.
Suggested correction: I don't believe it is ever established whether Arnold is left or right-handed in the series. Without confirmation; he may very well be part of the ambidextrous population that uses both. I used to write with both hands myself in school.
I do believe he is ambidextrous. In "The longest Monday" he is using chop sticks with his right hand. Many other episodes he uses his left hand to write.
Except the mistake is saying he switches hands in between shots during the test. Not that one episode he's left handed and the next he's right handed. Yes, the teacher was shown for about 1-2 seconds before the switch happened. But these mistakes should be considered valid as an actual mistake in animation rather than to suggest to the audience that for no reason whatsoever, Arnold switches hands to finish the test, and only does so when not seen.
I'm ambidextrous myself, and I often switched hands when taking tests. It was very useful for preventing hand cramps. It only takes a couple of seconds to switch the pencil to the other hand, so I don't see why Arnold can't do it within that span of time too.
Arnold's Hat/Stoop Kid - S1-E3
Audio problem: When Harold says, "Hey, what are you, scared?" his mouth stays open for the last word. It never closes for the "d" at the end.
Tutoring Torvald / Gerald Comes Over - S1-E13
Continuity mistake: Arnold and Gerald have to climb stairs to get to Oskar's apartment, but this would mean the building has three floors, and the outside views indicate that there are only two (plus the attic).
Continuity mistake: The first time you see the line of girls, Rhonda's shoes are "low-cut", but just before Helga says "Move it, football head", the shoes suddenly extend much higher (like they normally do in the stories).
Crabby Author / Rich Kid - S3-E17
Continuity mistake: In "Rich Kid" Lorenzo is introduced as a new student, but he was in a previous episode titled "Arnold's Room."
Suggested correction: Other than the few episodes where a specific time of year is taking place; these episodes are not meant to be taken to be shown in chronological order. Just roughly in the same 4th grade school year.
Suggested correction: I agree with the other correction, but it's also worth noting that in the production order, "Rich Kid" was made before "Arnold's Room." They were just aired out of order, just like many of the other episodes.

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Other mistake: Hall Monitor: Helga tries to convince Phoebe that being hall monitor is the greatest job, with plenty of perks. Just as Helga says, "I couldn't let this job go to anyone; I picked you," Helga's hand is on Phoebe's shoulder, and part of Phoebe's shoulder is missing. (00:03:10)

Continuity mistake: At the start, Lila is sitting on a bench near the school wall. However, in the shot where Sheena says, "Oh, that's wonderful," Lila and the bench are missing; then they reappear a moment later. (00:01:13)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Other mistake: Hall Monitor: When Helga chooses Phoebe to be the new hall monitor, as Phoebe starts walking to the stage, she is briefly layered behind the boy sitting behind her, and she is layered over Joey and the boy sitting next to him. (00:02:31)

Grandpa's Packard / Phoebe's Little Problem - S5-E19
Other mistake: Phoebe's Little Problem: When Arnold goes into Phoebe's room to cheer her up, the birdcage is reflected in the mirror on the wall, but the birdcage is too far to the right of the mirror, so it shouldn't be reflected. This error is shown again in a later shot where Phoebe tells Arnold, "And I'm very good at checkers." (00:09:13 - 00:09:45)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Other mistake: Hall Monitor: During detention, the class helps Phoebe realise how much she has changed. When Helga wipes the window so Phoebe can see her reflection, in the first shot, Phoebe's reflection is not backwards as it should be. (00:10:45)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Other mistake: Hall Monitor: At the start of the school assembly, when Principal Wartz thanks Helga for her month as hall monitor, Wartz's right arm is atop the lectern, with his fingers over the lectern's front edge. As Wartz gestures toward Helga with his right arm and hand, the fingers of his right hand oddly remain over the lectern's front edge. Then, after Helga announces she may volunteer again, the audience groans, and in the next shot facing Wartz, he has four hands at the lectern's edge. (00:01:35 - 00:02:00)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: After Phoebe's eyeglasses are stepped on, when she tells Helga, "Out with the old Phoebe, in with the new," Phoebe's belt buckle and two brown pouches have vanished, then reappear in the next shot. Later, while Phoebe sits on the desk during detention, her belt buckle is missing in her close-ups, and it reappears mid-shot as she crosses her legs. Then, when Phoebe's classmates share how they miss the "old" Phoebe, her buckle and the two brown pouches vanish, but they all reappear in a moment. (00:05:05)
Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: In the first and last overhead shots of the detention classroom, the "fire" sign, the fire extinguisher, and light switch are not on the wall beside the door, though all are on the wall in other shots during this scene. (00:09:30)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: While Phoebe is hall monitor, her watch repeatedly disappears and reappears on her wrist, even mid-shot. (00:05:00)

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: When Principal Wartz tells Helga she cannot volunteer again, he removes the "revered badge" and its black crossbody strap from Helga. Helga then selects Phoebe as the next hall monitor. When Phoebe joins them on the stage, Helga is again wearing the crossbody strap with the badge in the wide shot, but it's gone in the next shot as Wartz places it on Phoebe. (00:02:10)
Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: During the school assembly, Principal Wartz's necktie has diagonal stripes, and its top knot stripes switch to the opposite direction, then revert to the proper direction.

Hall Monitor / Harold's Bar Mitzvah - S2-E15
Continuity mistake: Hall Monitor: While Helga is hall monitor, her watch repeatedly disappears and reappears on her wrist.

Sid the Vampire Slayer / Big Sis - S5-E3
Other mistake: Sid the Vampire Slayer: While Sid stands in front of the mirror and points his rubber sword at Arnold and Stinky, the two ghoulish vampires, Arnold asks him, "How can we be vampires if we have reflections?" So when Sid is aware of his mistake, Arnold's eyes disappear and reappear in his reflection mid-shot.