My guess (so not a spoiler) - all of the people have done evil things and the Devil is coming to get them.
Can't he just claim them after they die? That's the usual method. Does he have to intervene because otherwise they'll somehow escape him?

Seriously, what would be the point?
They could become good, if given the chance to live. Or, if you're into that sort of thing, "form a personal relationship with Christ" and all that nonsense.
I'm embarrassed. Those are supposed to be the beliefs I subscribe to, and I didn't think about that! Too busy being snarky and cynical, I guess.
Although you'd think being confronted with the Devil would be the quickest way possible to convince someone to sincerely pray to Jesus.
Ah yes, the old "foxhole conversion."
Really though, I think I'm still trying to figure out what the nature of Hell and the Devil are supposed to be in an essentially atheistic Hollywood. Faith or salvation don't seem to play much part in it. Here we have a Devil who's apparently allowed to manifest himself in an elevator and... do whatever it is he's going to do. So why doesn't he just hang around permanently? I'd bet money this movie doesn't show a God or Jesus that's stopping him. And I'd bet double that anyone in the movie who actually does pray to Jesus will be shown up as a buffoon, or die a horrible death, or both.
In the Hellraiser movies, you've got a Hell that doesn't appear to be a normal afterlife at all, but it's still populated by immortal beings who traffic in immortal souls. And they'll take anyone they can get their hands ... err, chains on, not just the condemned. Does that mean they have the power to waylay someone from the afterlife they were meant for?
In Drag Me To Hell, you've got a gypsy woman with the power to curse someone to the traditional hell, in retaliation for a comparatively minor offense. So let me get this straight: she can exercise eternal judgement over someone else's soul for no real cause? How is it that she's not ruling the world?
I wouldn't presume to ask what the rules are in these movies, or get someone to explain the inconsistencies. I'm just trying to understand the mindset and world-view that comes up with this stuff.