The episode "Laughing Bat" where the Joker decides to become Batman. Just the general description is hilarious and watching as the Joker acts like Batman for the entire episode just puts a smile on your face.
When Joker-as-Batman shows up to foil his crime, Penguin gets ticked off.
"Hey, do I interfere with your 'gas the city' plots? No! It's called professional courtesy."
"There was a slight misprint on your invitations, sir. You are apparently hosting 'a farty'" (Alfred's completely deadpan delivery is what makes it so damn funny).
Batman has summoned his Batbot mech suit to take on a Venom-enhanced Joker. He's knocked out before he can get inside, so Batgirl does it instead. Batman awakes a few minutes later to see the battle, and says "She didn't" in a hilarious tone of chagrin and disbelief.
Later in the same episode, a Mini Mecha walking around like it's drunk.
Penguin is afraid of the mysterious Rumor that's been kidnapping all the supervillains in Gotham, so he's decided to turn himself over to Batman. So desperate is he to get caught that he resorts to the most cliched villain dialogue in the book.
Penguin: What I want is to turn myself in! Come on, chop chop! Let's go, pat me down! (Silence. Batman and Robin look at each other.) Have to catch me in the act, is that it? Okay, look! (Takes a painting off the wall) I'm stealing! (Batman, Robin, and Catwoman blankly stare at each other) What? Not enough? (Halfheartedly wields his umbrella) Then you should know that soon all of Gotham will suffer some cruel, ironic, bird-related fate at the hands of El Penguino! (Catwoman rubs her temple and shakes her head) What's a felon gotta pull to get arrested around here?
A lot of Robin and Batgirl moments turn into this. For example, when they're bored on stakeout duty:
Batgirl: ...Eaten by a crow.
Robin: I think he fell on a spoon.
(Viewpoint switches. They're staring at a wanted poster of a man with an eyepatch)
Batgirl: Nah, I bet he's not even missing an eye. He just thinks eyepatches are cool.
And then they debate over scones while Alfred tinkers in the background.
Robin: Anyone we know could be a robot!
Batgirl: Oh, it wouldn't fool us if we knew the person. ...Would it?
Robin: Maybe not. They'd probably be a little robot-y... detached... unemotional... mechanical...
(Both of them turn to look at Alfred)
Batgirl: ...Kick his head off.
Robin:You kick his head off.
Alfred: Attempt to do any such thing, and I assure you, there will be no hot buttered raisin scones for either of you for a year.
Robin: ...Only Alfred sounds that much like Alfred.
Robin's fighting an evil duplicate, who's copying everything he's saying:
From the same episode, when Harley sics Jokers hyenas on the TV crew.
Harley: So, you didn't like my show? Well, try this one. "When Animals Attack People I HATE!" (beat) It's a comedy!
In "Cash For Toys", Cash is sleepily walking through the house in the middle of the night when Alfred takes the elevator up from the Batcave and gets caught.
Cash (stares sleepily): Sorry. (Stumbles on)
The Penguin with Green Lantern's power ring. Enough said.
"A Fistful of Felt" has quite a few, such as Scarface's thug Rhino having to disguise himself as a goofy-looking smiling flower and Scarface (disguised as Mr. Snoots, a replacement puppet that was given to the Ventriloquist) threatening to destroy a kid because he asked him for a hug.