I think I understand fairly well by now why one wouldn't want tap water to get in contact with an electric circuit as it is conductive.
However, assuming the circuit will dry out completely before it is turned on, are there any risks for the circuit associated with having some tap water splash on it while it is powered off?
Best Answer
First of all: "turned off" != unpowered. There's batteries and large caps on many boards, and only luck determines whether these discharge in a non-damaging way when the board comes into contact with water.
Many PCBs are washed in distilled water after assembly, to get rid of flux and glue residue. Most components are designed to withstand that (of course, not all; I'd be a bit careful when it comes to things that are chemical sensors, or contain MEMS, which includes quite a few oscillators).
Point really is washed and distilled: you don't want to slosh around any salt or other soluble components on the board to have them accumulate wherever the water dries last, just for them to later, in high-humidity situations, temperature deltas etc, attract water from the atmosphere and become conductive patches.
Now, realistically: if your water was drinking water, it was probably clean enough that you will not have to care about that under realistic circumstances, so you'd probably be fine.
Another thing is plain old corrosion: surfaces get scratched during usage, and if, for example, that scratch removed the solder mask film from a copper trace, that trace will then oxidize and change characteristics. Water+freshly assembled PCB might have a different effect than water+used PCB.
All in all, in most cases you'll be fine, as long as you'd consider the water the board was exposed to clean, and as long as you really made sure to dry it thoroughly.
If the water wasn't clean, buy distilled water (car shops have that, but so do drug stores, …) and thoroughly "slosh" with that so that any dirt gets solved, then rinse with more distilled water. Let dry. Seeing you're from Porto Alegre, it might be a bit cold/humid right now, using an electric oven (not gas stove) with a built-in air circulator to dry your board at ~70 °C – 80 °C for an hour or two might be a reliable thing (don't make it much higher – plastic connectors tend to melt, and don't use the grill setting, if your oven has such).