I'm extremely skeptical of @leonardo's answer. I suspect that what would happen if you drank only distilled water is nothing perceptible. The only place where concentrations of distilled water would ever be high enough to conceivably matter is in the tissues of the mouth and throat, and even there, the effect would be temporary.
Compare drinking 8 glasses of either distilled or tap water every day. With tap water, you're looking at less than 200 ppm of Mg, Na, K, and Ca combined. That's less than 400mg of total mineral content per day. Given that the combined RDA of all of those minerals is on the order of 7g for an adult male, this is not nothing, but it's certainly small. Your dietary intake of these minerals probably varies by more than this daily, and your intestines, kidneys, sweat glands and mineral storage organs (like your bones and muscles) are constantly maintaining the mineral blood levels within a very narrow range, despite handling a throughput of several pounds of water and food daily. They might have to work slightly harder to manage this range if you drank nothing but distilled water, but in a healthy adult, normal intakes already vary by more than this amount without major problems. For example, the average American consumes more than 3.4 grams of sodium daily, while a low-sodium diet is on the order of 2g. Low-sodium diets have been widely studied in the medical literature, and are considered safe.
As for pH, the lowered pH is caused by increased carbon-dioxide absorption to form carbonic acid. Just as carbon dioxide is more soluble in distilled water, it is less soluble in stomach acid, and may be burped out. Would you die of acidosis from drinking seltzer water all the time? If that were the case, I'm sure there would be big health warnings about drinking soda, while it seems relatively benign. Furthermore, your body produces and excretes (through the lungs) around 1kg of CO2 daily, dwarfing any extra CO2 you might get by drinking distilled water. If the small amounts of CO2 found in distilled water were dangerous, jogging would be invariably fatal.
If you were to drink nothing but distilled water, and eat no food, you probably would die of hyponatremia within a few weeks. But you would also die of hyponatremia if you were to drink nothing but tap water, though perhaps slightly more slowly.