Hey hey! Time's up, folks. Here are being the answers:
1: But my mom says I'm cool.
Milhouse, Burns' Heir. Burns specifically requested no nerds!
2: That monkey is going to pay.
Professor Frink, 22 Short Films About Springfield. Frink shows up too late in the episode to get his segment because of an accident at the lab.
3: I've said it before and I'll say it again: democracy simply doesn't work.
Kent Brockman, Bart's Comet. Congress is about to approve a bill to bail out Springfield's population, but the measure is shot down after a congressman adds a rider that would set aside $30 million "for the perverted arts."
4: You'd better run, egg!
Homer, Homer the Great. Stonecutters episode. One of those Egg Council guys got to Lenny, too.
5: Hello. I am not interested in buying your house, but I would like to use your restroom, flip through your magazines, rearrange your carefully shelved items and handle your food products in an unsanitary manner. Ha! Now you know how it feels!
Apu, You Only Move Twice. AKA the episode with Hank Scorpio.
6: My eyes! The goggles, they do nothing!
Rainier Wolfcastle, Radioactive Man. Almost everyone got this. Context is obvious is you recall the episode at all.
7: Rocky five plus Rocky two equals...Rocky VII, Adrian's Revenge!
Bart, Lemon of Troy. Yep, lemon tree episode, Bart is on the run from Shelbyvillians, etc.
8: Don't cry for me, I'm already dead.
Barney, A Star is Burns. Film festival episode, line is from Barney's movie.
9: "Sex Cauldron?" I thought they closed that place down!
Krusty, Grade School Confidential. AKA, the one where Skinner and Krabappel start dating (Krusty still fails at spelling). Number of people cited the burlesque house episode for this one.
10: Well, if it isn't my old friend Mr. McGreg, a leg for an arm and an arm for a leg.
Dr. Nick, Homer's Triple Bypass. Title is self-explanatory, no?
11: You want the truth? You can't handle the truth. No truth-handler, you. I deride your truth-handling abilities!
Sideshow Bob, Sideshow Bob Roberts. Bob rigs the election when he runs for mayor, etc.
12: I am the lizard queen!
Lisa, Selma's Choice. Selma takes Bart and Lisa to Duff Gardens, Lisa drinks the...not-water.
13: What IS your fascination with Daddy's Forbidden Closet of Mystery?
Chief Wiggum, This Little Wiggy. Bart is forced to endure a "play date" with Ralph, winds up getting some riot gear out of it.
14: My Homer is not a communist. He may be a liar, a pig, an idiot, a communist, but he is not a porn star!
Grandpa, Simpson Tide. Homer enters the navy, screws up as expected. "It's my first day." Also see Rozalia's avatar.
15: Thank the Lor -- thank the Lord? That sounded like a prayer. A prayer. A prayer in a public school! God has no place within these walls, just like facts have no place within organized religion.
Supernintendo Chalmers (most of you actually called him that. This makes me very happy), Sweet Seymour Skinner's Badasssss Song. AKA, the hundredth episode. Skinner gets fired and replaced with Ned Flanders.
16: Tastes like...burning!
Ralph, Das Bus. Lord of the Flies episode (Ralph eated the purple berries!)
17: Don't touch my stuff! ...Heeey, this isn't the YMCA.
Lionel Hutz, Marge on the Lam. Marge and the new neighbor go all Thelma & Louise, Lionel Hutz is appointed babysitter.
18: And so concludes our tale. I'm Leonard Nimoy. Good night, and keep watching the skis. Er, skies.
The squeaky-voiced teen (or the acne-scarred or pimple-faced teen, labels vary), reading from the script in The Springfield Files (alien sighting turns out to be Mr. Burns). While Lenoard Nimoy is in the episode, he ditches the narration two-thirds of the way through and joins the rest of the cast in the story. I did say I wasn't using celebrity cameos, right?
19: "You know, I don't think I ever heard the word 'embiggens' before coming to Springfield."
"I don't see why not. It's a perfectly cromulent word."
Ms. Krabappel and Ms. Hoover, Lisa the Iconoclast. Lisa discovers Jebediah Springfield was a pirate.
20: You kids might remember me from such educational films as ``Lead Paint, Delicious But Deadly'' and ``Here Comes the Metric System!''
Troy McClure, Bart's Friend Falls in Love. AKA, the one where Milhouse gets a girlfriend. McClure shows up in a sex ed video Bart's class watches, vowing to approach the matter "in a frank and straightforward manner," and then showing a cartoon about rabbits with muttonchop sideburns.
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Scores!
Soppy: 38
NEB: 33
Trips: 25
CK the younger: 23
Meeple: 23
Doma: 21
Zenny: 20
Fudozukushi: 19
Shale: 17
CK the elder: 12
Hats off to Soppy. The man knows his Simpsons (note: I gave him bonus points for pointing out the origin of Purple Monkey Dishwasher, even though I hadn't intended that as a question).