If I peeled and diced potatoes and put them in cold water, changing the water frequently, do they still have to be refrigerated? If they are not refrigerated will they go bad? 

Once a potato is cut its usable shelf life clock begins to accelerate dramatically, versus keeping the potato intact and properly stored. You could constantly change out cold water on diced potatoes as you describe, for a very short period. However, the product will eventually begin to discolor and this does not take the place of refrigeration. Whenever we field a question about food safety my advice is always to never tempt fate. Prepare and cook the potatoes as usual, and avoid taking any chances that could possibly compromise your health and safety.