For starters, there's Asuka's introduction in episode 8. Touji's hat gets blown off by the wind. Asuka (who's wearing a yellow sun dress) steps on it, and Touji spends a while trying to get the hat until it lands at Asuka's feet and the wind blows Asuka's skirt upwards. Cue the title card for the episode, with the normal sound replaced by the sound of Asuka slapping him (as well as Shinji and Kensuke, who had the bad luck of being right behind Touji when this happened). After the title card, Touji decides that the best thing to do would be to return the favor but he lands up using too much force and pulling his underwear off as well. He gets bitchslapped so many times by the horrified Asuka that for the rest of the episode his face is beet red. Despite not sounding very funny, it's actually utterly hilarious in the show. Here's the first part of the episode. Asuka's facial expression of bug-eyed horror in that scene is priceless.
Touji: Oh, I think it's overpriced! That's okay though, here's your change!
Later on, there's the lunch scene where Kaji asks Shinji if Misato is still so "wild in bed". Following that, there's the scene with the inappropriately small elevator. Notice that in this scene, Shinji's face is wedged between Misato's boobs.
Misato & Asuka: Hey, watch where you're touching! Kaji & Touji: I can't help it!
Following that, Asuka forcing Shinji to wear her spare plugsuit is always good for a laugh, and then there's the start-up sequence for Unit 02, during which Shinji accidentally screws things up with his "thought noise." Asuka tells him to think in German if he has to, so Shinji lists off "Strudel...Bratwurst..." An exasperated Asuka switches the system language to Japanese.
Also "Strudel...bratwurst..." "DUMMKOPF!"
All the more hilarious if you watch it in Japanese and you're German—the only thing he can come up with is "Baumkuchen", which is a cake available basically everywhere in Japan all year round, but comparatively rare in Germany outside of Christmas season. The German version of this scene manages to be confusing and hilarious at the same time—confusing because the whole scene is in German, so the joke does not quite come across; hilarious as Shinji's attempt to "think in German" is "Schnitzel...Pumpernickel..."
In a blink and you miss it moment, the episode also includes a close-up of Misato's ID card. It includes fields for age, weight and measurements. She's scribbled over all the fields with a magic marker.
Don't forget the episode where Shinji and Asuka had to train together to defeat an Angel in perfect synchronization. After the battle, Asuka starts yelling at Shinji for the incident that had just happened the night before the fight, Shinji is all "NO U" and everyone in the NERV HQ (Professor Fuyutsuki included) is either facefaulting or laughing their heads off... LOL, Shinji and Asuka. Just LOL.
The absolute best part was the aftermath of their first battle. The enemy splits in half, and we get a hard cut back to HQ with Asuka and Shinji being chewed out over the battle... complete with photos of their EVAs stuck upside down in the ground with the legs sticking straight up.
There's also Ritsuko Akagi's news commentary on the aftermath: "This ispathetic."
Fuyutsuki commenting on the use of N2 weaponry on the Angel: "We'll have to re-draw the map again!"
Another highlight is when Toji and Kensuke bump into Hikari on the way to visit Shinji and Asuka respectively. It's only when they're right outside the door that they realise they're going to the same place. Then when Asuka and Shinji answer the door, wearing matching outfits and talking in synch, the visitors get the wrong idea entirely. Capped off by this exchange:
Hikari: That's filthy, you two!
Shinji/Asuka: You're wrong!
Hikari: Two wrongs don't make a right!
This is followed by Misato and Rei visiting. Toji jerks his finger at Shinji and Asuka and asks what is going on while Hikari and Kensuke continue the same horrified poses.
Also, in episode 11, when NERV's air conditioning isn't working because of the power failure, and everyone is amazed Gendo and Fuyutsuki don't seem to be affected by the heat. Turns out they definitely are (The view cuts to show Gendo's feet. He's taken off his shoes and socks and has his feet in a fire bucket filled with cold water to keep him cooled), and their whole conversation on the subject is as follows:
In episode 5, Pen-Pen hesitantly samples Misato's atrocious cooking and promptly faints.
Everything about that dinner scene. Everything.
One time when Toji and Kensuke come to pick Shinji up, I remember a conversation going something like this:
Asuka: I'm getting dressed for school, and if you peep, you're DEAD!
Toji: Who'd want to watch a snake shed its skin?
The wacky alternate universe sequence in episode 26. Schoolkid antics, morning stiffies, Misato the wacky schoolteacher, and Gendou getting scolded for being lazy.
Rei as a Genki Girl takes the cake. She is seen running late for class, toast in her mouth, before crashing into Shinji and inadvertently flashing her undies at him. Shinji and Touji discuss this much to Asuka's annoyance, then Rei shows up as a new student. All hell breaks loose, with Asuka accusing her of being a tart and Rei retorting that she's riding Shinji's baloney pony.
In the English dub of this, there's this exchange:
Asuka: Hurry up!
Shinji: Sheesh, you're such a pain in the ass sometimes. Is this what happens when you start your period?
Asuka: WHAAT?! WHAT DID YOU SAY?! [very audible slapping noise]
Misato is riding the elevator which stops on Kaji running down the hall. Misato calmly reaches over to hit the button to close without batting an eye. Then Kaji makes it in time and her face scrunches up in a disgusted scowl.
A nonstandard moment of funny from ''End'', where there really shouldn't be anything to laugh about. After Instrumentality, there's that sea of LCL scene with Shinji and Rei in the water, naked of course, like many people in this series often are, and she's straddling him in such a way that... Anyway! What's hilarious about this scene is that it seems like the staff throwing their hands up in the air over all the Misaimed Fandom surrounding the two and exclaiming something like: "You like undertones? We'll give you damned overtones to make you shut up and leave us alone!" It gets even funnier when Kaworu magically shows up, as being the only one clothed it gives the impression that he just walked in on the two of them, or wants in. Or something else equally Freudian.
I snickered at the first live action clip showing an empty movie theatre. I don't know what the intention was behind it, but to me it seemed to say: "Okay, we're well into the mind screw section of the movie now. Next, we have - hey, where'd everyone go?"
The entire Alternate Universe where everyone's just living a normal life in episode 26. Doubly so in the English dub, which saw fit to randomly give southern accents to Misato and Rei.
And then there's "Evangelion Alternative Silly Ending" or, as it's become known in the fandom, "Shinji's Rant".
Shinji: "Okay, okay, okay. I mustn't run away, I mustn't... Okay I got that, good, okay. Now, if I were to run away, let's analyze that—where the fuck would I go?! I'm on a big blue ball!!"
When Shinji runs out of the shower. Thanks to the wonderful rules of perspective, we get to see him bolt into the kitchen, where sits Misato and a very convenient beer can on the table to censor Shinji's... well. Not only does this scene drag on from this perspective for at least thirty seconds of speaking, but then Misato picks up her beer, only to reveal a small can of toothpicks behind that.
The toothpicks thing just seems to be either the writer or animator's way of adding insult to injury for poor Shinji.
Another good one is during a semi-public bath scene in a spring divided by a wall—Misato and Asuka are bathing on one side, and Shinji and Pen-Pen are on the other. Shinji is treated to a lot of suggestive-sounding dialogue from the girls. Viewers can only see his blushing face for a long moment, and when the camera pulls back, the penguin looks down at Shinji's crotch, covered by the water, as if something interesting is going on.
It has to be mentioned, the infamous Commentary Of Evil, where the English VA crew take the incredibly dark End Of Evangelion and turn it into a comedy. How much a person likes it is usually inversely proportional to how much one likes the film itself.
Hey wait, that last guy's still alive. * blam* Oh, no, wait, yeah, that's all of them.
On the subject of Eo E, if you watch the 'everyone turning into tang' section out of context it is highly amusing to see the SEELE guy say "Everything is going according to plan" and then promptly melt.
Later in the original anime (ep. 17), there's this exchange:
Kaji (to Shinji, after flirting with Misato and Bridge Bunny Maya): How about having a cup of tea with me?
Shinji: I'm a guy.
Scene is topped by Rebuild's version, in which Kaji decides to freak Shinji out and moves just a little too close. Shinji's scream, which is is probably what makes the entire scene, is audible clear down the hall.
When Shinji goes to Rei's apartment, and then he meets for the first time... after she had just gotten out of the shower. Shinji is extremely nervous, Rei walks up to him, and then he trips and falls onto her. After a fairly long moment of silence, Rei says "Will you get off", which causes Shinji to freak out even more because he realized that his hands were on her breasts.
Shinji Ikari Raising Project changes this scene to Rei giving a cold smile, before bitch slapping him to the ground. Doubles as Crowning Moment Of Awesome for anyone who wanted to see this done to the Butt Monkey, and triples as a bit of a Running Gag as Rei slaps him again after his comments about his father.
I recently watched the English dub of NGE and I don't remember which episode it was (it was that one episode where it's simply a still shot of a mostly empty room with dialogue between Misato and Kaji), but one bit of dialogue in particular was almost rib-breakingly funny.
Misato: Don't stick weird things in there!
One from episode 1, where Shinji enters Unit 01's hangar for the first time. As they enter, the lights switch off, causing Shinji to scream and yell "Why's it so dark in here?!" Then the lights come on again, and Shinji spots Unit 01, screaming again as it's face-to-face with him.
Kaworu's No Sense of Personal Space in the manga leads to a rather funny scene where Kaworu barges into Shinji's shower stall looking for soap, then wonders why Shinji's freaking out.
The following exchange in episode 16.
Makoto Hyuga: You look pretty worn to the bone, Major Katsuragi.
Misato Katsuragi: I have reasons for being worn. Private ones.
The entire Evangelion: After The End radio drama, in which the cast of Evangelion is offered a chance to resurrect the show and keep trying to retool it in order to secure better ratings. Possibly the funniest moment is Asuka's idea to retool Evangelion as a sentai show. Then there's the ending, where due to having no budget and only a few minutes left till airtime, the voice actors decide to play out the first episode of Evangelion by themselves, along with all the sound effects and music... which results with five minutes of black screen with the voice actors making cricket and missile noises, with a cappella "The Beast" playing in the background, which then ends with Misato going "EVA-01 just defeated an Angel!".
And it works!
Rei suddenly turning into a Motor Mouth and slapping the tar out of Asuka. Words cannot describe the hilarity.
"'Anta baka, anta baka!' You're the baka!"
Pen-Pen also suddenly talking, also a Motor Mouth, and Asuka pretty much doing the same thing to shut him up!
Even better is that in this audio drama, Asuka's complaining that the newly redesigned plugsuits are too revealing, mentioning specifically that they show her panties completely and that her nips are clearly visible and it's even obvious what colour they are. When the "Test" plugsuits were unveiled in Rebuild of Evangelion, they were oddly familiar...
Episode 9 contains several jokes for the purpose of Asuka's Character Development, each of them quite funny. This includes:
In the English dub, Asuka's shout of "Scheist!" when Shinji messes up the dance routine, before kicking him so hard, he flies into the air for a few seconds.
Misato asking Rei to do the dance routine with Shinji, and they manage to replicate it perfectly, to Asuka's dismay.
The montage showing the two competing over everything, from the dance machine to changing TV channels to brushing teeth.
And the scene where Shinji is having a Potty Emergency and Asuka, accidentally, slams the bathroom door in his face. Made funnier by Pen-Pen taking the opportunity and running in.
The final argument at the end (also in the English dub, though it is funny in Japanese as well), first over Eva-01 landing on Eva-02, then over the kiss the two nearly shared, done over a hologram. Somehow, Asuka actually manages to hit Shinji over the head, despite just being a projection. On top of this, they are standing in a crater they created while killing Israfael, nonchalantly having a childish argument like nothing happened. While this is happening, all of the NERV staff can see and hear them. The looks on their faces are priceless. Even Kozo Futsuyuki, of all people, facepalms.
Misato: Aww, geez!
Episode 7 includes a scene in which Shinji berates Misato for being slobby, adding, "I can understand why you're still single at your age." Cue Misato trying to calmly respond with the most fake smile you have ever seen (her eyebrow is actually twitching with rage). It doesn't sound funny, but it is when you're watching it.
Gendo: I've given the old men of SEELE something else to look at. There is no problem.
[cut to a naked Ritsuko standing in front of SEELE]
In Chapter 34 of the manga, Shinji wonders why Hikari keeps asking him about Toji when she constantly yells at him, and Asuka responds that it obviously shows she likes him, and that he'll never understand "the female psyche". Shinji then starts wondering if being harsh is how women show they like men, thinking about Misato and how sheacts aroundKaji. He then starts considering what Asuka's harsh behavior towards him might mean, and Asuka's reaction is hilarious.
Shinji: (thinking) Wait! Asuka is harsh to me...No way! That's just how she is-contrary...Mr. Kaji's lucky.
In the manga, Shinji tries to see if Rei can laugh by doing something funny - in this case, shoving two fingers up Touji's nose, causing him to freak out. This makes all the other girls laugh, but not Rei, who merely turns away with the same blank expression after seeing what all the noise was about.
The image makes it look like she's shaking her head at the scene. Somehow that makes it funnier.