What are the effects of storing drinking water in a copper container?
As you state, copper storage containers have long been used to prevent growth of pathogens in potable water. 

As for the danger of copper toxicity, it depends on the pH and water temperature. Between pH 8 and 11, about 1 ppm of copper dissolves, so if you're consuming ~5 l/day or about 5,000 g/day of water, and if the water has had sufficient time to become saturated, then you'd get ~5 mg/day of copper, exceeding suggested safe limits for long-term exposure.

That said, I think that, if the water pH is between 8 and 11, and if it is refrigerated (e.g. 4 C), and if it has been standing quiescently for no more than eight hours, it would not have time to be saturated with copper, so it probably would be safe to drink if that were the only source of copper in your diet. 

Given all those mealy-mouthed caveats, then, you might at least check the pH of the tap water, and use a different container if it's acid. You would not want to be like the drinkers from Yudono spring, unless you're really into Heavy Metal.