This is hard to answer with an authoritative source since the quality of non-potable water varies widely from the source it comes from (is it groundwater? river water? is there anything recycled from drainwater from houses?) and even within a country.
I'm speaking mostly from my experience living in India, but in practice I have seen similar practices in Malaysia, Indonesia, Vietnam and Thailand too. None of these places have potable water supply; the source is typically river or lake water. The reason why they are classed non-potable is primarily because of high mineral content or particulate matter, which can be purified for human consumption by physical filtration as opposed to bacterial disinfection - although as it has become cheaper and more compact to make ultraviolet or reverse osmosis filters for home use, these are also used. Bacteria or viruses are the causes for most water-borne diseases, those can get into the water supply if untreated raw sewage is dumped into water sources which thankfully most civil authorities are not stupid enough to do.
So for the countries I mentioned, in practice, while it is customary to purify water for drinking either by boiling or using filtration devices, for other purposes such as washing clothes, washing dishes, cleaning hands, bathing, brushing teeth et al - essentially, anything that does not involve ingesting water, tap water is used directly. The water may taste funny for instance if you're brushing your teeth - that's because of the high mineral or iron content typically - but as for safety, yes, it can be considered safe.