I was awake for ages last night thinking about this. Can water go off, like food and other drinks do?
If you poured a glass of tap water and just left it sitting on the side for X amount of time, would it grow bacteria that would make you ill if you drank it?
Yes I think so . That why you get stagnant water ponds. I think it would be more likely with bottled water as tap water has chlorine and flouride in it (well in some areas) which kills bacteria.
Tap water contains virtually no bacteria and none which causes illness - so, if covered, would remain safe. Far safer than bottled water which has been partially drunk - this introduces bacteria from your mouth which then multiply - how quickly depending on temperature.
...oh, and if the tap water is river-derived as opposed to underground sourced, it may well go slightly green because microscopic algae will grow in the light. However, still safe to drink.
If you leave water in a kettle overnight, and then use it again next day, that doesn't taste too good, though probably safe. I usually water indoor plants with it, before filling afresh