It's a huge challenge to figure out what to cut out and what to leave in -- and it's perhaps a bigger challenge to remove something, once you decide it doesn't belong. I always tell my students, "A sentence must earn the right to live on the page." But that also applies to storylines and characters. In fact, it's a great test, asking yourself, "If I remove this (word/sentence/plotline/character) from the story, will it affect the story in a major way?" If the answer is clearly "no," then that element should likely be cut.
And next, I tell my students, "Be strong! Be brutal!" when it comes to eliminating something you know should be cut out. Because, in the end, it will make the story better, stronger. And the end result should always be to serve the story.
I saw a great article today by "Television Without Pity" (just like the name sounds, these articles drip with biting sarcasm). It lists the Top 10 Characters Whose Absence Would Greatly Improve Their Shows.
Granted, they seem to choose their criteria based on whether or not they simply like a character. And though that should be a factor for a writer, I think there's more at play in making that all-important "to stay or not to stay" decision. Things like how would his/her absence affect other characters? Or how would it affect plots and sub-plots? Would his/her absence make any difference at all?
Here's a good example: While editing my novel last year, I decided to "cut out" my protagonist's mother. She'd been an extremely minor character who didn't really serve the story in any way. And something happened when I "cut her out" (I actually killed her off): my protagonist became a little deeper. Now, instead of having an annoying, nagging mother whose phone calls she begrudgingly took once a week, my protagonist became a little lost, a little empty, because she'd lost her mother a few years before. There was a wound now that wasn't there before. And that made her even more interesting.
In the end, the decision is the writer's. But if we have a gut feeling about a character -- that he should be cut -- and we just can't bring ourselves let him go, I think the story will suffer. Perhaps not overtly, perhaps not terminally. But there will forever be something nagging at the reader (and the writer) when the story is read.