Discounting mechanics, I would divide the effects of magic items on the plot into 5 main categories: Aesthetic, Intelligent Items, MacGuffins, Curses, and Sledghammers.
The best example of an Aesthetic affect of a magic item is the classic flaming sword. Sometimes its fun to swing around something that flashes and sizzles. Plot wise though, you're not missing much.
Intelligent items are basically just another NPC. Talking swords, hats that take a person over, and so on.
MacGuffins are the goal: You need that one magic arrow to slay the dragon, or the mystic sword of greatitude to declare your kingship, and so on.
Curses are simply fun: Wear the ring and the wraiths come to kill you. Wear the hat and get lice. That ring will turn your skin green. And you can't take any of it off!
Sledgehammers are magic items designed to solve a particular situation. Think about the Zelda games: you can't pass this area until you find those one gloves that let you move that one block, or the gadget that lets you jump over the chasm, and so on.
I'm not sure any of these are irreplaceable with a bit of thought.