Is it necessary to refrigerate containers of bottled water after opening?

Bottled water under normal conditions should not require refrigeration after it has been opened. But there is the possibility of it becoming contaminated after opening and support growth of microorganisms, since there is nothing added to the water (like chlorine) to prevent such growth.

Perhaps you can find more specific information here: http://www.cfsan.fda.gov/~ear/retfood.html#water

If you drink from the bottle, yes. our mouth has bacteria in in that can contaminate the bottle. If you pour it into another container, then no.