There's one cutscene where Mario sets off the detonator after beating Roy Koopa in the fifth castle, only to see the castle not blow up. He goes to take a look at it... and then the castle blows up right in his face.
In a similar vein, after beating Ludwig von Koopa in the fourth castle, Mario sets off the detonator to blow up the castle... only for the castle to launch into the air like a rocket and impact against a nearby hill... which then wears a bandage where the castle hit for the rest of the game.
"Mama Luigi" provides several:
Luigi: Well, like they say in Brooklyn, early to bed, early to catch the worm...Or is it the bagel?
Yoshi: Mmmmm. Worm's good.
Yoshi becomes attached to Luigi:
Baby Yoshi:[jumps into Luigi's arms and licks his cheek] Mama Luigi!
Luigi: Cool it, caterpillar breath! I'm not youró
Mario: Mama? Mama Luigi?! AHAHAHAHAHA!
After Yoshi eats Mario:
Mario: Now I know how a meatball feels!
Baby Yoshi:[spits Mario out] Ueagh, Yoshi no like meatballs!
"Fire Sale": Kootie Pie asks Mama Fireplant if she's been putting on weight.
At the end, Kootie Pie becomes so cold, she cries ice cubes.
"The Wheel Thing":
Kooky: Fasten your seat belt, it's gonna be a bumpy ride! Koopa: What seat belt?
Koopa: Any more bright ideas? Kooky: Uh... fasten your seat belt?
"Send in the Clown":
Koopa: A circus isn't a circus without a dinosaur-taming act! Mario: Uh, how about a chipmunk-taming act instead?
Also, Big Mouth being terrorized by the clowns during the performance.