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Most Buffy and Angel-centric episodes are bound to have cheesy dialogue, with optional Fauxlosophic Narration.
Angel: It's not the demon in me that needs killing, Buffy. IT'S THE MAN!
Any time the show did giant snakes, like Lurconis or the Spawn of Sobek. The sad thing is that they did giant snakes so often because, by the FX team's own admission, giant snakes were easier than other options (with Lurconis in particular, with a giant rat, all you can do for scale is a foot or nose sticking out of something; with snakes, it's easy to imply there's a lot more off-camera).
The utterly ridiculous ease with which some Mooks and Monsters of the Week are killed. Watch as one of the vampires in "School Hard" seems to purposely swerve into the path of a cart kicked in his general direction.
Season 1
There's the scene in "Angel" right after Buffy finds out that Angel is a vampire, in which Buffy throws Angel out her living room window. While her mom was home.
In "The Puppet Show", it looked like the enemy would be a talking puppet. (It turned out not to be the enemy, but there was still a talking puppet! And it was horny!)
Fans just love to mock the scene in "Prophecy Girl" when Angel tells Xander that he can't perform CPR on Buffy because, being a vampire, he has no breath. Except that they had been running down a tunnel for a while, and David Boreanaz is audibly winded as he says the line.
Season 2
In the otherwise extremely effective episode "Passion", Angelus kills Jenny Calendar and burns her computer to destroy her findings on how to restore his soul... except, he didn't on the latter. Angelus only destroys Jenny's monitor, leaving her computer—the actual important part that holds all her files—completely untouched. Willow could have easily recovered Jenny's findings by simply buying a new, working computer monitor and hooking it up to Jenny's system. This was, of course, before computers were a staple in most households and knowledge about how they worked wasn't common, so this detail likely slipped by audiences at the time. However, to a modern audience watching back, Angelus' blunder is pretty narmy.
Season 3
There is a scene in "The Wish" after Vampire Willow and Xander die, and the Master and Buffy are literally pushing people from one side to the other in an effort to get to each other. Dramatic to the point of funny, the scene probably defines narm.
The demon from "Gingerbread" moves slowly, but somehow is fast enough to accidentally stab itself through the throat. Buffy doesn't even move the giant stake towards it - she just points it the right way and lets the demon jog forward to its death.
The Hellhounds that attack The Prom look like monsters when they attack people and/or move short distances. When they move long distances, it's so obvious that they're humans in suits who have to hop like rabbits because they can't really run on four legs.
In "Bargaining Part 2", Xander tries to reassure Willow about the urn of Osiris. It's been broken before they could finish the resurrection spell (or so they think) and there's some sad music playing. Xander suggests patching it together with duct tape and crazy glue... completely serious, as if he genuinely thinks that will work.
When Buffy is turned invisible in "Gone", she tries to talk to Dawn, who shrieks out the quite inexplicable line "How am I supposed to talk to you when I can't see you?" One wonders how this girl handles talking on the phone.
Dark magic gets you high, as in you literally float up to the ceiling. Fantastic.
At the start of "Villains", Willow randomly summons the god Osiris just by screaming for him, seemingly just so the writers can have a god informing the audience that Tara's death can't be magically undone. While this might be an attempt to call back to the urn of Osiris used to resurrect Buffy, it still comes wildly out of left field, Osiris is played by the floating head of a man in cheap makeup, and Willow banishes him afterwards by screaming.
Dark!Willow's "bored now". It was supposed to be horrifying and a reminder of the awesome creepiness that was Vampire Willow. With Miss Hannigan playing Dark!Willow like she was sleepwalking, it didn't really have the intended effect.
Dawn is an attention-seeking kleptomaniac, Giles walks out, Xander leaves Anya at the altar because he's afraid they'll end up like his parents, Anya sleeps with Spike after she returns to being a demon, Spike takes off after trying to rape Buffy, Buffy is near-fatally shot, and Tara is killed immediately after reuniting with Willow, who had just kicked her addiction to dark magic. All of these twists and drama bombs occur in such rapid succession that looking back, it's almost hilarious, in the same way an overwrought soap opera is. By the time the finale rolls around, the angst levels have gotten so ridiculous that Giles bursts out laughing after he's filled in on everything he missed that year, and Buffy promptly follows suit.
Season 7
For Canadian viewers only, Spike's trigger in Season 7, since it's the same tune as the old kids' show The Friendly Giant. Imagine if the First turned Spike evil by singing "It's a beautiful day in the neighbourhood" or "Sunny Days, Sweeping the Clouds Away..." and you'll have an idea of how Canadians saw that scene.
Throughout the episode "Help", the characters spend large chunks of time reading Cassie's poetry. To the show's credit, the poetry does read like something a teenager would write. To the show's detriment, it tries to pretend laughably bad teenage poetry is deep and insightful.
From "Showtime" there's Andrew saying "two men enter, one man leaves" as Buffy begins to fight the Turok-Han. It Makes Sense in Context, but Andrew says it right in the middle of a big dramatic confrontation and it comes across as genuinely serious, rather than the frequent nerdy references Andrew is known for. It seems like the show is trying to play it straight rather than using it for parody.
In "First Date", after Buffy and Xander get back from their dates and people start joking about Xander's penchant for hooking up with demon women, Giles gets angry at everybody and chides them quite loudly about their lack of seriousness. Then he uses the flash cards he made for Chao-Ahn to make his point.
