Batman at a wharf trying to get rid of the Cartoon Bomb, finding nowhere to dispose of it without running into things like a Salvation Army marching band, a woman pushing a baby carriage, nuns, and even a family of ducks. Batman in frustration:
"Some days you just can't get rid of a bomb!"
This recently became utterly hilarious in hindsight given that The Dark Knight Rises recycled this particular scene as its climax.
At the very end after the Security Council is re-hydrated. The way West delivers those lines are just plain HILARIOUS.
It gets better. Right before they leave, Robin and Batman glance back, watch the Security Counsel bicker in their new languages, and then turn away in a manner that suggests "Yes, we are definitely done with this. Let's leave, but inconspicuously." And then they leave. Out the window.
During the big fight on the submarine, Catwoman decides to throw her cat at Batman. Bats catches it and runs around with it, trying not to get hit, and finally tosses the kitty into a life raft and gets back to the brawl.
Batman: Bon voyage, pussy.
BAT.SHARK REPELLENT. Even better, the shark explodes immediately afterward (not that it makes any more sense, but the shark had a bomb in it).
Selected from an assortment of sea-life repellents, including Whale Repellent.
While fighting off the baddies out-of-costume, Bruce Wayne knocks one goon onto a spring-loaded catapult and sends him flying out the window (as a stiff dummy) and into the sea. The goon then explodes (The trap, intended for Batman, catapulted the goon into an Exploding Octopus which the Penguin had prepared offscreen. Again, not that this makes any more sense).
The Penguin trying to trick his way into the Batcave as Commodore Schmidlap. How well does his disguise work?
Best part is Penguin knows they know, but has an ace up his sleeve in the form of dehydrated henchmen. Unfortunately, he had no way of knowing that the Batcave uses soft water, which causes the henchmen to instantly disappear with a cartoon sound upon being struck. Then Batman goes into a short speech about the value and fragility of life. Bathos at its finest.
Batman: [reads the second riddle] What people are always in a hurry?
Robin: Rushing people... Russians!
Batman: So this means...
Robin: Banana... Russian... Someone Russian is going to slip on a banana and break their neck!
Batman: Precisely, Robin! The only possible meaning!
The final line is really what makes this the best of them all! One of the only films ever where the heroes exclusively use Insane Troll Logic.....and it works.
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.
A couple times, Catwoman randomly lets out a long, drawn out and enthusiastic "meow" and both times, one of the villains' henchmen give her a total "WTF" look.
Batman's sizing-up of the implications of the Villain Team-Up. He claims any two of the four would only team up if their plan was to take over the city. Three would only team up to take over the country. But all four? They're trying to Take Over the World. At minimum.
When Batman & Robin are magnetically stuck to the buoy and the torpedoes are heading towards them, Batman is using a transmitter to blow the up prematurely. Then it stops working.