Movie intro text:
"Sometime in the 23rd century
the survivors of war, overpopulation and pollution
are living in a great domed city, sealed away from the
forgotten world outside. Here, in an ecologically balanced world,
mankind live only for pleasure,
freed by the servo-mechanisms which provide everything.
There's just one catch:
Life must end at thirty unless reborn in the fiery ritual of carrousel."

Year of the novel (published): 1967 (1+9+6+7=23)
Year of the movie (release): 1976 (1+9+7+6=23, 2+3=5, Logan 5, 2274 (22+74=32, 3+2=5 & 32=6))
Month+day of the movie (release): June 23rd (6+23=29, And Logan is 29, 29-23=6, Jessica 6 (76), 2274 (2+2+7+4=15, 1+5=6 & 15=5))
FRANCIS: Runner terminated at zero 16. Ready for clean-up Cathedral quadrant 39. (39-16=23)
016, 1+6=7, 39, the numbers 3, 7, 9 (379), 23 (23), 3792*3=2274
The Carousel computer announcements were something dubbed-in at final assembly. And possibly the writer of that particular line, "Year of the city 2274" was on the same page as you, interesting thought. The vintage advertisements plus other promotional material, even the alternate titles of the film in several countries indicates the audiences' 23rd Century (After Death).
Vintage poster:
http://www.cinemasterpieces.com/82011/jdp453.jpg

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One thing I wondered was in the film it says it is the year of the city 2274 does that mean its the year 2274 AD or when humanity moved into the domes they started back from 0 and humans have been living in the dome city for 2274 years

to blend the BOOK ideas with this movie. in the book, the youth were tired of the really bad decisions and laws made my older government, and nuked Washington DC. Then they created a system using a master computer controlling everything, and decided to eliminate everyone once they reached the age of 21: party harder until then, then snuffed out.
So, this movie is sort of a POST book look, and instead of the whole world being this way, it was just this protected city.
The connection to the book is kinda messy and it's best to not worry about it.
But the YOUTH and comptuer controlled part are the same. Mostly.

Pretty sure they don't say and it's better that way. None of the characters would even be able to understand it anyway. This way, people can project their own fears for the end of civilization as the cause. Your father grew up in the Cold war, future kids will probably believe it due to global warming. Maybe they just decided it was for the best and there was no catastrophe!