Any idea why tap water goes stale overnight?
Exclude reasons one by one
Surface biological contamination: use two vessels, wipe the one of them with H2O2, hydrogen peroxide solution, leave another untreated. Check if time till smell appears is different by a few days. Bacterias can survive 20% vinegar and 100*C for a short time, some of them, if they are in a sleeping stage, it is not quite reliable method i would say.
Tap water biological contamination: use two vessels with water, add a few tablespoons H2O2 with water in one, and just water in other. Check if time till smell appears is different by a few days.
Tap water chemical contamination: buy some bottled water, use two vessels, one with tap water, another with bottled water. Check if it will be different in a few days.
Bulk material of the vessel releasing chemicals: leave water in different material vessels, one is glass, another one is your preferred one. Check if glass vessel has less smell adter a few days. Glass releases the least amount of chemicals into the water.
Dust accumulation: use two vessels, cover one and leave another one open, check if smell will appear faster in one.
Psychological factor: use two identical vessels, ask your friend to give you one or another, randomly, blindfold yourself, do a test 10 times, check if your score is above 8 correct guesses out of 10 tries.
After completing this tests you will narrow down your search significantly to give you some idea about what is going on.