Chico asks Big Boss to talk like Trenya, the talking cat.
"Uh... Meow...? Meow... Meow meow meow..."
"Wow... I have no idea what you're saying!"
Cecile trying to speak to Trenya is just as hilarious.
Chico trying to grab at Amanda's cig (read: leaping as high as he can... while Amanda holds it juuuust out of reach).
It seems that almost any time Snake meets a new character, he introduces them to the Fulton Recovery System in the best way possible, resulting in them screaming as they're yanked off into the sky. Although it's not shown, it's implied that he did this to Huey, who is in a wheelchair. Most of the characters comment about it in their first conversation afterwards - Amanda and Chico's are hilarious.
The unlocking of the ability to recruit volunteer soldiers begins with Miller suggesting to Snake that "recruiting" soldiers by violently kidnapping them through Fulton might not be the best way to go about it now that they're a bit more respectable.
If you read the letter for Strangelove, you get to hear Huey pissed off over Snake reading the letter meant for Strangelove.
Quite a few of the briefing files are like this. Highlights include Snake and Miller talking about cardboard boxes, Snake talking to Huey about NORAD, and any time Miller's trying to get Cecile's attention.
"He's real I tell ya! He used bring me presents and..."
The whole Santa Claus scenario is kind of heartwarming in a way. It's a good way to remind the player that, before the events of Metal Gear Solid 3, Snake was essentially a giant child. One can't help but picture a small Big Boss excitedly opening his presents on Christmas morning...
If you Fulton up soldiers when Mother Base is at its population limit, Kaz calls you surrounded by crowded soldiers. "Hey, stop pushing!"
"Damnit! Man overboard!"
Paz's Diary, Entry Six.
If you're fighting a Monster Hunter boss, call Cecile: "Here, let me try and talk to it. -ahem- RAWR! RAAWWWRR!"
Losing Extra Ops 67: Date With Paz. "Creep!" -stamps away- "Paz, wait! I'M SORRY! I DIDN'T MEAN TO- Paz! PAAAAAAZ!!!!!!!!" Rarely, various characters will sing the game over theme if you lose the mission in your underwear.
The Human Slingshot. May be Awesome, but Impractical due to needing a full four-man team and LOTS Of grinding to pull it off. This is all naturally overshadowed by the fact that you can launch a human at a tank and potentially blow it up.
Bothering either Paz or Kaz when they're knocked out during the dates results in some humorous codec criticisms.
The call before the date with Kaz.
Kaz: "Snake...the sunsets are so pretty here..."
Snake: "Are you feeling OK Kaz?"
There are missions where you have to hold up all the soldiers without being seen or it's instant game over. Turns out that's because you're armed with nothing but a banana
The dates with Paz/Kaz are deliciously video-gamey, you have to woo your company by using the combat centered co-op commands to hilarious results.
Snake: "BOX TIME!"
Huey's combat rating is an 'E'. That means Huey has the capability of a low level soldier. Let that sink in for a moment.
The short section involving searching trucks for the nuclear warheads is loaded with easter eggs in many of the trucks. Such as a truck loaded with metal gears (resulting in Snake's famous response), or Hideo Kojima hiding in the back of one. And you even recruit him as a merc!
The various unique cardboard boxes, including the Box Tanks (which include a functioning explosive or stun grenade turret!) and the ambulance box, which even has a rotating red light on the top.
The cutscene right before the final battle, Paz goes on and on about Cipher's plan and how BB and MSF will be their tools, before threatening to use ZEKE to kill you, the same ZEKE you can unequip everything but the main parts and railgun from right before the battle. This troper just found it hilarious that she thought the whole monologging would do anything, also that at the end she describes the events of Operation Intrude N313 perfectly, 25 years before it happens.