From the same episode, the scene when the residents are moaning about the furnace breaking:
Grandpa: You call this cold? You should have been here in the winter of '49!
Mr. Hyunh: I don't care, I was in Vietnam in 1949...plus I was not born yet!
"Eugene's Bike": Arnold and Eugene enjoying a day out, and all of the bad things happening to Eugene (chokes on a hot dog, gets hit in the eye with the piece that flew out of his mouth, falls into the river while looking out over the docks, etc).
The episode "Heat" has a number of them.
Grandma: I can't stand it Arnold! It's hotter than blazes! I'm a grandma with a husband, a young grandson, and a boarding house full of people to be responsible for.
Arnold:[to himself] This heat's so crazy it's got Grandma acting normal.
The entire scene with the Jolly Olly man.
Helga:(climbs atop ice cream truck) We've had enough! We want our ice cream, and we want it at a price we kids can afford!
Harold: You said it, sister!
Iggy: That's right!
Helga: And if we don't get it, we're going to do something really bad!
(Crowd yells in agreement and rocks truck.)
Jolly Olly Man: Okay, okay! (thinks) You can have "Rum Raisin" for 10 [dollars].
Eugene, trying to help (in his typical clueless way) by attempting to lift up the entire back end of the ice cream truck. He predictably hurts himself, and spends the rest of the scene marching back and forth and clutching his back while the other kids rock and shake the truck.
In "Wheezin' Ed":
Arnold: Who'd be stupid enough to make counterfeit pennies?
And the moment shortly thereafter when they run into the counterfeiters:
Gerard: Eleven cents?! Boy howdy that's completely outrageous! It's unacceptable!
Kid: Quit overreacting Gerard and cool out.
Arnold's reaction to Lulu flirtatiously winking at him: "Oh please."
After Helga's confession in The Movie, and an awkward kiss, Arnold states he's dizzy, and needs to go lie down. Helga's response: "Wonderful...I'll go with you!"
And let us please not forget Arnold's Grandma. "Hey! Crazy lady! Come back here with my bulldozer!!"
From "Phoebe Cheats", Mr. Simmons reads Helga's love poem for Arnold for a poetry contest in class and its signed Anonymous. When she hears Mr. Simmons begin to read it she freaks out and slouches down in her chair prompting the following line from Stinky:
Stinky: Gosh Helga you're sliding down your chair like it was your poem or something.
The Running Gag of Helga slugging Brainy in the face whenever he breathes down her neck. Another instance has him randomly lay his head in her lap. She responds by punching him. Also, in one episode when, instead of punching his lights out, she finally asked why in the world he's always lurking behind her:
Helga: Look, Brainy, this is just weird. How is it that you're standing behind me again? How did you get in this little arch? Were you waiting for me to come to this alley? What's your deal?
It's made funnier by his expression: "She's not punching me. What the heck is going on?!"
One time, Helga just walked away without punching him, causing Brainy to suffer a panic attack and then punch himself out.
Anytime Eugene has an Epic Fail. Then there's this scene where he gets attacked by chickens. Look at Arnold in that scene, he couldn't care less!
Mr. Hyunh: Why do you always have to cheat, Oskar! Why can't you act like a normal person?
Oskar: Me? What about you? You're wearing a dress!
Mr. Hyunh: That is to help Arnold! I'm Juliet!
Oskar: You're not Juliet. You're a sad man wearing a dress!
On the episode where Helga forges Olga's report card (making her think she got a B and sending the poor girl into a depression) and begins getting all the gifts that Olga turned down (as she thinks she's a failure just because she got a B), Helga eats from a box of chocolates and gloats how she finally took over being the favorite of the family — until she chokes on the chocolate and has to give herself the Heimlich maneuver.
The episode where they try to get into the Book of Records by making the world's biggest pizza puff.
Gerald: I hope we put in enough baking soda.
Phoebe: 150 teaspoons should be plenty.
Sid:(nervously) "Teaspoons"? I didn't know it was... "teaspoons."
The entirety of "Suspended", especially Harold's repeated attempts to break into the school.
Mr. Redman's robot butler Mr. Egg Cream mistaking Grandpa Phil for Redman and offering him egg cream.
Helga does some pretty spot on impressions of her classmates, which offends them. Phoebe suggests she do something more positive, so she writes her a REALLY CORNY poem complimenting her classmates in PAINFUL PAINFUL rhyme. It's not that funny, but the music that plays while she's reading it never fails to make me laugh.
Helga: "Curly, now there's an original kid." *weird rimshot noise*
The episode where Arnold's grandpa thinks he's going to die.
Grandpa: I must be in Heaven now. (opens his eyes, sees Oskar) Oh no, Oskar's here! I must be in the other place!
In the episode where Phil goes back to school, Arnold asks him where he's going in the evening with two sixth grade girls.
Phil: We're gonna sneak into a rated PG-13 movie.
Arnold: But you're 81! You can get into a PG-13 movie anytime!
When the teachers go on strike, Principal Wartz has to teach the entire school. He tries to lead everyone in a sing-along...
Principal Wartz: And now, the left side of the room! B-I-N-G-O...
Another episode has Helga give herself a makeover, in order to make herself look more girly for Rhonda's party. When her mother opens the bathroom door and sees the finished product she gasps and faints, causing Helga to say "Maybe I should've gone a bit lighter on the eye shadow."
Susie yelling at Oskar for not helping out around the house, especially this line.
Susie: "You expect everyone to take care of you."
Oskar: "I don't expect everyone to take care of me, just you."
Goose: [Beat] ...Curly! Go stand in that corner facing inward! [Helga laughs] You've earned yourself a corner, little lady! [Harold laughs] And so have you, pally! Move, move, move! [Points at Stinky] You! The remaining corner now!
In one episode, Principal Wartz is trying to get Arnold to state who mooned him (Wartz), but Arnold won't because he's not a snitch. Wartz tells a story of how in school he tattled on some classmates and could Never Live It Down, essentially destroying his social life. He states that he didn't care what anyone else thought of him, because he knew he had done the right thing. And in the end, he became an elementary school principal.
Wartz: ...Wouldn't you like to be an elementary school principal?
Arnold:(Unimpressed) Not really.
(Wartz's eyes pop open with a comical 'boing!')
In the episode "Field Trip" where Arnold and his grandma break into the aquarium to free an old aged mistreated turtle, they pass by a security guard who is too preoccupied with tormenting the animals to even notice them. Though what he does is actually pretty funny
Guard:(laughing) Here have a cookie. (penguin coughs up cookies and squawks angrily) Uh oh, you tossed your cookies. Here have a cookie. (penguin coughs up cookie and squawks angrily) Uh oh, you tossed your cookies again.
Later on when Arnold and his grandma are escaping with said turtle, the pass by the guard who is now taunting a shark.
Guard:(dipping his hand in and out of the shark tank) Let's go big shark. Come on, Jaws. Bite me. (falls into the tank and is cornered by shark) WHAAA!