Logan was running to avoid being arrested for theft...
Sorry for the sensationalist heading, but some might find this interesting.
Spoiler warning
The movie and I presume the book 'Logan's Run' is probably stolen from a short story that was then stolen for a radio play, while the concept of every person getting killed at 30 is probably stolen from an Isaac Asimov novel that was turned into a radio play.
I realize that it's not uncommon for two (or more) people to have the same idea at the same time independently (Leibniz and Newton and their simultaneous independent discovery of calculus, for instance) but since both of the ideas were presented in radio plays, I can't help but think this is a little too coincidental.
The Asimov novel and the Dimension X radio play are both called "Pebble in the Sky"
https://www.youtube.com/watch?v=re4Fca7orBQ
In this novel and radio play, the age of death is 60, and it is referred to as 'the 60.'
The X-Minus One Radio Play is called "The Sense of Wonder" and it was apparently stolen from a Robert Heinlein short story named "Universe."
https://www.youtube.com/watch?v=QSQAB0wfrcU
I haven't read "Universe" but Logan's Run takes all sorts of plot points from "The Sense of Wonder" including:
1.The action taking place in a living space where the equivalent of a city population is confined (a spaceship in this case)
2.The people being confined to this area due to an apocalyptic event
3.A man meeting up with a woman in the course of the action
4.The man discovering that there is a much bigger world (universe) than what they are confined to
5.The man and the woman being on the run from authorities for having this knowledge.
6.In the case of the radio play the spaceship luckily happens to land back on earth as this action is occurring. Just as in Logan's Run, the man causes the confined living space to be destroyed to force everybody into the wider world.
This radio play is clearly meant to be a critique of dogmas, especially religious dogmas, but like most of these critiques, the radio play itself can be easily critiqued as a rather un-entertaining polemic.
I can't comment on any similarity here to Logan's Run, because for me the movie made no sense. The idea behind everybody dying at 30 is 'one new person comes in and one old person goes out.' (I don't remember if that's the exact quote.) If the people wanted to 'run' and leave the city rather than get killed upon turning 30, why did the authorities (or the Sandmen) care? All that matters is that they left.
I read here the film was significantly edited, so maybe the original version explained this, but without an explanation for the reason for the Sandmen and preventing people from leaving the city, it's impossible to know if the original movie version (I haven't read the book Logan's Run) also was a critique of dogmas or not.
Anyway, agree or disagree with my critique of Logan's Run, but fans of the movie (and the book) might find these radio shows and books interesting as well.