Potatoes - Cold refrigerators cause potato starches to be turned into sugar. This extra sugar causes them to darken prematurely when cooked and makes them oddly sweet - not in a good way. The Idaho Potato company recommends that you store potatoes in a cool, dark place. Doing so will keep them fresh for 1-3 weeks.
Tomatoes (OK, it is a fruit) - have a bad reaction when in the cold: they get soft and mushy. The cold causes the membranes in the cell walls to become damaged giving you an overall mealy texture. Instead, store tomatoes at room temperature, out of the sunlight (which can ripen the fruit unevenly), and stem side up for minimal bruising.
Onions - become soft in the fridge -- and what's an onion without its crisp bite? They also have the tendency to impart their flavor on surrounding produce. And while you might love the taste of onions, you probably don't want everything to taste like one. It's actually better to keep whole onions in a warm dry place until you're ready to use them. Once cut, you should store them in the fridge, covered; they will keep for a few days.
One more thing to keep in mind: never store your onions and potatoes together. Onions and potatoes can both give off gases that will cause the other to spoil faster.