In one of the Riddler's appearances, he is seen toting around a sack full of ill-gotten gains which is helpfully labeled "LOOT SACK".
The constant labeling of things in general. Batman mentioning that Penguin must have the supplies for a specific heist in his "criminal storehouse". Cut to Penguin (listening in via bugged umbrella) sending his minions to fetch said supplies...from a room labeled "CRIMINAL STOREHOUSE".
Batman at a wharf trying to get rid of a Cartoon Bomb, finding nowhere to dispose of it without running into things like a marching band, a woman pushing a baby carriage, nuns, and even a family of ducks. Batman in frustration:
"Somedays you just can't get rid of a bomb!"
Batman and Robin deploy a gadget called the Bat Ladder which is like any normal deployable ladder but with a sign saying "BAT LADDER" at the end of it.
Jack Nicholson's Joker has plenty of his own funny moments, but his reaction to the Batwing whisking away the Smilex gas-dispensing balloons is priceless: "HE STOLE MY BALLOONS! Why didn't somebody tell me he had one of those... things?" He promptly asks his right-hand man for a gun — and shoots him.
Joker: Bob. Gun. *bang!* I'm gonna need a moment alone, boys.
Batman: Pardon me, have you ever danced with a devil in the pale moonlight? *punch* *bong*
and, during Batman's fight with the goons, one of them tries to jump him, only to fall through the floor, perfectly in time with the music.
This dialogue between Lt. Eckhart and Knox.
Knox: You know what they say? They say he can't be killed. They say he drinks blood. They say—. Eckhart: I say: you're full of shit, Knox. You can quote me on that. (flicks his cigar at him)
"Are you deaf? You don't speak English?"
"Mr. Knox, we have enough problems in the city without worrying about ghosts and goblins."
"That's not a denial!"
"Check this out. He must have been King of the Wicker People."
"Actually, he's Japanese." "How do you know that?" "...Because I bought it in Japan."
Bonus points for Bruce being behind them the whole time they were looking.
And for Robert Wuhl, who ad-libbed his musings on Bruce's collection.
Batman's smile, as if saying "thanks".
Joker's meeting with the crimelords.
Joker:[electrocuting Antoine Rotelli with his lethal handbuzzer] Oh, I got a live one here.
Joker: Antoine's got a little hot under the collar. Vinnie: You're crazy. Joker: Have you ever heard of the healing power of laughter? (laughs while wiping his "skin" off to reveal a pale, clownlike skin underneath the forehead.) NOW GET OUTTA HERE!!
"I'm only laughing on the outside. My smile is just skin deep. As you can see inside, I'm really crying. You might join me for a weep."
The Joker's surprise present for Vicki.
"Now you fellas have said some pretty mean things, some of which were true, under the fiend Boss Grissom. He was a thief and a terrorist. On the other hand, he had a tremendous singing voice."
"Where is the Batman? HE'S AT HOME! WASHING HIS TIGHTS!"
Joker's dance with Vicki on top of the old church. And his "BANG!" gun.
Joker: Here, let me lend you a hand. [Vicki grabs Joker's hand, which is a fake, causing Vicki to fall. But Batman saves her.] HA HA HA! "Lend you a hand"!
"They don't make em like they use to! Hey gang? Hey Batsy?"
The way Joker tips over the little statue at the museum.
At the start of Bruce's story to Joker:
Bruce: Lemme tell you about this guy I know, Jack... mean kid, bad seed. Hurt people.
Joker: I like him already.
Bruce: You know what the problem was? He got sloppy. You know, crazy. He's the kind of guy who couldn't hear a train until it was two feet from him.
And this clever one:
Joker: Bruce... Wayne, n'est-ce pas? ("isn't it so?")
Bruce: Most of the time.
While Vicki was arguing with Bruce:
Vicki: I trusted you! I even slept with you, I can't believe I did that! Bruce:(rolls his eyes) Well...
Alicia meets the Joker for the first time. "Honey, you'll never believe what happened to me today!"
Joker dropping in on Grissom:
Grissom: That you, sugarbumps? (sees silhouette of a man) Who the hell are you?
Joker: It's me...sugarbumps.
In the Black and White comics, there is a short story set in World War II era Gotham. Batman catches Catwoman stealing diamonds and is highly doubtful when she claims her victim is a Nazi spy. They stake out the man for a while out of mutual stubbornness.
Batman: I haven't seen anything to prove that claim is true— Victim: *saluting his colleagues* Heil Hitler!
The Comically Serious: Batman is always so serious, it is INCREDIBLY easy to make anything funny by having him do it in a dramatic manner. It's so true that the comically serious page has him as the trope page.