What’s the name of the movie with Peter Cushing as a veddy sniffy, veddy proper English scientist, and he’s with a bunch of other people, and they’re in a cave or a tunnel or something, and they have this huge drill machine they call the Mole – and the last shot is of the Mole drilling its way up through the White House lawn!
At the Earth’s Core. Might want to take a look also at these two other ERBs adaptations : The Land the Time Forgot and its sequel The People that Time Forgot
Just on the cusp of the '70s (but still feels like it and is worth a viewing) is 1980’s is Lathe of Heaven.
There was Heaven’s Gate – oh, wait, you didn’t want movies that were disasters. 
This is the one I remember making the most impact from my youth.
Strangely, the other ones I want to mention (The Birds, Earthquake, something with a spinning tunnel), are all from a trip to LA’s Universal Studios as a kid.
I think I came up with the ultimate in another thread:
Nazi Dragodinocrocosharktopus
I’m childishly peeved that **The Omega Man **has already been suggested. So in its place, I suggest Zardoz.
Though possibly if you’ve seen one film starring Sean Connery running about in a post apocalyptic world wearing thigh high boots and what looks like a red diaper, you’ve seen 'em all.
A Thief in the Night and A Distant Thunder
A semi-professional filmmaker throws together amateur actors & scrapes together whatever sets & props he can to make B-grade Left Behind/Rapture/AntiChrist films- not to make big bucks off best-selling novels but because he & the cast & crew believe in their guts that this is true & people need to be warned.
They have this raw primal energy that make them glorious!
I didn’t think Zardoz was all that bad. Yes, the costumes were silly. The ‘chanting’ thing at lunch was early-‘70s spiritual-crystal-gazing-woo-woo, and annoying. But I saw it as a teen around 1980, and it had naked breasts. It had a legitimate story, but it was done rather tongue-in-cheek; like something a UCAL student would make. I like the line, ‘Absolute acquittal,’ and the dialogue ‘We’ve been used!’ ‘And re-used!’ ‘And abused!’ ‘And amused!’ I read in Starlog that the ‘ship’, a giant head’ was built for a production of The Lord Of The Rings that was never made. (The magazine had an article on space ships – which also introduced me to Dark Star.) If you look at it expecting it to be serious science fiction, then yeah; it’s laughable. But if you look at it as somewhat comedic science fiction, it works.
I haven’t seen The Omega Man since the '80s. I remember liking it.
I have just looked through my collection and will add a few others…
Clockwork Orange
Colossus: The Forbin Project
Executive Action (not exactly post-apocalyptic, but pre-supposing it could go there)
The Boys from Brazil
Alien (1979, baby!)
Deathrace 2000
Fahrenheit 451 (late '60s)
THX 1138
The Last Man on Earth (also late '60s)
Arguably a dystopian story, but in no sense a disaster/armageddon movie.
The book (The Hephaestus Plague) was better. 
The original **The Crazies **by George A Romero (1973). Might as well throw Dawn of the dead (1978) in there as well.
Some other disaster/armageddon/dystopia films:
Deathsport (1978)
Wizards (1977) - Heavily rotoscoped animated film by Ralph Bakshi.
Zardoz (1974) - Sean Connery rides a giant stone head into a civilization of imortals
Invasion of the Body Snatchers (1978)
Mad Max (1979)
Soylent Green (1973) - It’s people
Westworld (1973)
Ebert’s review of this one (which is not on his website <!>) is a classic.
ETA: found it on Google Books. A passage:
OK, it’s from 1988, but I have to throw in with Miracle Mile starring Anthony Edwards, of all people. Starts off like a fluffy little RomCom, then blindsides you with a horrifying speculation on the lead-up to Armageddon. I saw it once. Once, and twenty years later several scenes still freak me right the hell out just thinking about them.
1989, but I’m going to recommend Blood of Heroes with Rutger Hauer in a dystoian future as a player in a bloody game reminiscent of rugby or football.
StG
Three words.
The Big Bus. It’s… an experience.
I’ll second that.