Star Trek 6 is really good.

Two exchanges I always liked not from ST6:

DS9, It’s Only A Paper Moon:

"When the war began… I wasn't happy or anything, but I was eager. I wanted to test myself. I wanted to prove I had what it took to be a soldier and I saw a lot of combat. I saw a lot of people get hurt. I saw a lot of people die. But I didn't think anything was going to happen to me. And then, suddenly Dr. Bashir is telling me he has to cut my leg off. I couldn't believe it. I still can't believe it. If I could get shot, if I could lose my leg, anything can happen to me, Vic. I could die tomorrow. I don't know if I'm ready to face that. If I stay here, at least I know what the future is going to be like."

"You stay here, you're going to die. Not all at once, but little by little.Eventually, you'll become as hollow as I am."

-Nog and Vic Fontaine

Voyager, Living Witness:

“ When diplomacy fails, there's only one alternative: violence. Force must be applied, without apology. It's the Starfleet way.”

-Janeway

Obviously very different reasons that I like each.

SNW S01E04

Context is important on this one, but even without the rest of the episode around it I still really like it.

Spock: "The lower decks are beginning to buckle."

Noonien-Singh, urgently: "The evacuation isn't complete there's still crew down there!"

Pike, with intensity: "Get them out."

Spock: "Decks are collapsing, if we do not seal them off now loss of pressure could spread to the whole ship."

Pike, pained resignation: "...Seal the bulkheads."

Noonien-Singh: "Structural collapse on deck 22, we've lost one crew person."

Spock, to Pike: "You made the logical choice..."

Pike, to Spock, quietly: "Why doesn't it feel like that?"

Spock, to Pike: "For the same reason you made it: because you value life."

This one from DS9: Rocks and Shoals has always stuck with me.

Sisko: And when they've reached this point, we'll have them in a crossfire.

Gordon: They won't have a chance.

Garak: That is the point. In case you've forgotten, we're in a war.

O'Brien: There are rules, Garak, even in a war.

Garak: Correction. Humans have rules in war. Rules that tend to make victory a little harder to achieve, in my opinion.

Gordon: So we just shoot them down?

Neeley: They wouldn't hesitate if the situation was reversed.

Nog: But we're not the Jem'Hadar. It is our duty to —

Sisko: THIS ISN'T A VOTE!

(stunned, everyone turns and looks at Sisko)

Sisko: The decision's mine. And Mister Garak is right. We are at war. Given the choice between us or them, THERE. IS. NO. CHOICE. Let's move out.

"What does God need with a starship?"

One of my favorites is from Sisko, but I guess this one is more of a soliloquy than a dialogue,

The trouble is Earth! On Earth, there is no poverty, no crime, no war. You look out the window of Starfleet Headquarters and you see paradise. Well it's easy to be a saint in paradise, but the Maquis do not live in paradise! Out there, in the Demilitarized Zone, all the problems haven't been solved yet! Out there, there are no saints! Just people! Angry, scared, determined people, who are going to do whatever it takes to survive, whether it meets with Federation approval or not!

Sulu: I'll protect you, fair maiden.

Uhura: Sorry, neither.

"Well double dumb ass on you!"

--Captain James T. Kirk

Seriously though there are so many scenes with Picard laying down some wisdom that it is difficult to pick one. I suppose I would go with:

"It is possible to commit no mistakes and still lose. That is not failure, that is life."

--Captain Jean Luc Picard

I would agree about the dinner scene. I also think the ST II genesis debate between Spock, Bones, and Kirk is great. I especially like Bones’ lines comparing God making the world in seven days to man’s hubris of the genesis device.

Some good examples here. I'd just mention a few other memorable dialogue scenes:

• all the scenes of McCoy with Spock's Katra in ST:III, including lines like "Yes, Genesis! How can you be deaf with ears like that?" and "Where's the logic in offering me a ride home, you idiot? If I wanted a ride home, would I be trying to charter a space flight?"
• in Star Trek IV, the scene with Kirk and Spock in Gillian's pickup, with Nimoy doing his best Gracie Allen impression. Also, the "No dipshit" bit in the restaurant.