The bran and the germ are high in nutrients and oils, and are thus more prone to spoiling and going rancid. Flours that contain these parts of the wheat grain have a shorter shelf life than flours that contain only the endosperm.
So what does this mean for storage?
All flours eventually go stale or spoil, and this date is usually indicated on the bag itself. Refrigeration would slow down the rate of spoilage, especially with whole grain flours. However, if you're a frequent baker, chances are that you're using the flour faster than it would go rancid anyway.
Our conclusion is that it's not a bad idea to store whole grain flours in the fridge, or to store all flour in the fridge if you don't bake very often. However, it's not 100% necessary and if space is a commodity in your refrigerator, keeping flour in your pantry is perfectly fine.
Regardless, it's best to buy only as much flour as you think you'll use within a year and then throw away any leftover flour after its expiration date. Expired flour won't necessarily make you sick, but it will give your baked goods off flavors.
Where do you keep your flour?
Related: How To: Organize Your Fridge