Han's conversation with Greedo while he's held at gunpoint, during while he slowly pulls his blaster out of its holster under the table and prepares to shoot him. His Karmic Death was totally hilarious!
Then as he walks away, he tosses a coin to the bartender and says "Sorry about the mess".
The Special Edition "Greedo shot first" cuts are just as much, either showing Greedo to have incredibly bad marksmanship that he misses Solo up close or showing Solo as dodging the shot in an animation that looks like a 2 second photoshop job.
Then Chewie pretends to freak out and attacks the officer, while Han shouts "It's loose!" and he and Luke act like they're going to subdue him but instead shoot the guards and the cameras in the room.
That scene where the stormtroopers unlock the door and start shuffling in. The leader is giving out instructions as they walk in but right behind him one of his men hits his headon the door frame with a notable *CLUNK*.
This was an accident during filming that they decided to keep and even gave a sound effect to.
A bit of Fridge Logic in "A New Hope" is that when Luke, dressed in full stormtrooper gear first meets Leia, she actually flirts with him. Does she make it a habit to flirt with imperial troops?
She might have been trying to present a nonchalant appearance after being tortured, as in "You think that probe was bad? You never went to Spring Break on Aldreaan!"
Alternatively, she was trying to get the trooper's guard down until she could take him out, steal his weapon, and fight her way out.
Or she was just being snarky. Telling a guy that he's short isn't generally a good way to flirt with him.
Or the entire sequence where Threepio forgets he's wearing a comlink, while Luke and the others are calling for help while trapped in a garbage chute; with the walls closing in. Then, when Threepio finally remembers and tells Artoo to shut all the chutes down, and then proceeds to confuse screams of jubilation for horrified death-rattles.
C3PO: Listen to them, they're dying, R2! Curse my metal body, I wasn't fast enough, it's all my fault!
Or any of Threepio's interactions with Artoo during A New Hope. The original intent for Threepio was for him to be a fast-talking car salesman, but it was derailed - arguably for the better - when Anthony Daniels (his actor) decided to play all of Threepio's car salesman lines completely earnestly, making him into a British gentleman. As a result, the writing shifts so that he became less snarky in the later movies.
C3PO: Just you reconsider playing that message for him! [R2 beeps a question] No, I don't think he likes you at all. [R2 beeps again] No, I don't like you either. [R2 beeps sadly]
Another good Threepio moment is the visual gag when R2 and Threepio are pulled out of the sand after the Sail Barge fight using magnets, and Threepio is upside-down.