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.
Commodore Schmidlapp casually reading a book on Penguin's sub, utterly oblivious to the destruction that's going on around him.
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 Schmidlapp. 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 accidentally rehydrated them with the heavy water (2H2O) used for the Batcave's atomic pile, 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.
And speaking of paper-thin disguises, earlier on, Batman goes on a date with Ms. Kitka. He makes Alfred tail them in the Batmobile (because Robin's too young to drive, of course). Alfred's crime-fighter costume is just a black eye mask over his regular clothes, with his glasses over his mask, making it hilariously obvious that he's just Bruce Wayne's butler in a mask. Driving Batman's car. No big deal. And, of course, if he should get pulled over, he had earlier confirmed to Bruce that he has his driver's license in his wallet.
Batman: Right! Two: "What people are always in a hurry?"
Robin: "Rushing" people...? Russians!
Batman: Right again. Now what would you say they mean?
Robin: Banana... Russian... I've got it! 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.
To be fair, all of those riddles were actual riddles back then and didn't originate with the show. There were whole books of elephant jokes and banana jokes and similar weirdness.
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 gives 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.
The opening has a few. It's shot in a series of dark alleys, with a spotlight illuminating the focal point. At first, it's a series of dedications written on brick walls. The spotlight turns on, then moves over to where the first of the dedications is written. It cuts and does it again for the second. Then on the third, it stops too soon, catches a couple in the middle of a heavy make out session, then double-times it past them to the third.
Then, for Adam West's casting call, the spotlight shines on Batman, who appears to be sneaking around. When he notices the light, he stands up straight, as if to say "Forget it...", and disappointedly shambles away.