Beginning: Pulp Fiction (with 1977’s Star Wars being a close runner up).
Ending: Without hesitation, The Godfather, Part II.
Beginning :Joe Vs. The Volcano
Beginning: Patton
Got it in one.
Beginnings
Sexy Beast
Sunset Boulevard
Touch of Evil
Endings
Casablanca
Song Is Born (the Judy Garland version)
Sunset Boulevard
The Usual Suspects
Dr. Strangelove
Beginnings:
2001: A Space Odyssey
Parenthood
The Rocky Horror Picture Show
Endings:
2001: A Space Odyssey
Dancer In The Dark
Grease
Revolutionary Road
Thelma & Louise
Most of those aren’t big iconic moments remembered by everyone like some others mentioned in this thread, but I think they open or close their film perfectly.
Beginning
The Lion King
The Sword of the Stranger
Dark Knight
Ending
Fight Club
Monty Python’s Life of Brian
I think the beginning of Contact needs to be on the list. It’s just enthralling.
The bit that plays during the credits for X-Men Origins: Wolverine is another good one.
Beginning:
Star Wars
Saving Private Ryan
Leon
Ending:
Seven
The Good, the Bad and the Ugly
Blade Runner
Some of my favorites, in no particular order:
Beginnings:
Up ~ Wordlessly eloquent
Dazed and Confused~ the parking lot, with Sweet Emotion playing
Inglourious Basterds ~ the interrogation
Endings
Hope and Glory ~ the school
*Places in the Heart *~ everyone at church
Local Hero ~ the phone ringing, and Mark Knopfler’s song
McCabe and Mrs. Miller ~ the final battle in the snow
Beginnings:
The Indiana Jones movies all have great openings, except for #4
The Fellowship of the Ring (the prologue with Sauron)
The Matrix (Trinity fighting the cops and then escaping the agents)
Speed (the elevator rescue)
Endings:
The Shawshank Redemption (the reveal of Andy’s escape and its aftermath, and then Red going to meet Andy on the beach)
It’s A Wonderful Life (“No man is a failure who has friends.”)
The Right Stuff (“But on that glorious day in May 1963, Gordo Cooper went higher, farther, and faster than any other American - 22 complete orbits around the world; he was the last American ever to go into space alone. And for a brief moment, Gordo Cooper became the greatest pilot anyone had ever seen.” Cue triumphant music)
Openings:
Citizen Kane
Butch Cassidy
Manhattan
Indiana Jones 1 & 2
The Godfather
Up
Watchmen
Avatar
A Touch of Evil
Saving Private Ryan
Endings
The Italian Job
Psycho
Citizen Kane
Indiana Jones 1
The Silence of the Lambs
Ending
Butch Cassidy and The Sundance Kid
The Usual Suspects
Lethal Weapon 2
I like how we the audience come in during a high speed chase that’s already in progress. No build up, just pedal to the metal from moment one. And it’s funny!
Kill Bill
David Carradine’s voiceover is hypnotic and scary. When the gunshot came, I jumped from my seat the first time I saw it. Then my ass sat back down and couldn’t move until it was time to pop in the sequel.
The kid in me will always have a soft spot for the opening of Star Wars. I was seven years old, sitting in a dark movie theater, looking at a strange world with two moons! Two moons! And then this huge spaceship flew into the picture from overhead, and I thought, WOW!
…and then this utterly enormous star destroyer flew into the picture from overhead, and I thought, …guh? 
Beginning:
Subway
Ending:
Withnail & I
I was 15 going on 8 when Star Wars hit the theaters. It was an older, single screen theater with lines around the block, we bought tickets to the next show and came back with two grocery bags full of pop and snacks. We waltzed right in with all that stuff because they were so overwhelmed with people that they didn’t have time to kick us out or take it from us. The whole experience was overwhelming, but that beginning? I don’t think there had ever been anything like it.
I like the beginning to Beetlejuice.
For endings, I keep thinking of melancholy ones, like* Late Marriage.*
Oh, and I agree Joe vs. the Volcano has a neat opening.
ETA: Oh, I remember now my favorite ending: Mad Max: Beyond Thunderdome.
A stunning beginning (if you don’t mind slow) that I don’t think was mentioned is Once Upon a Time in the West.
A nice ending I saw recently was The Third Man.