If I leave a glass of water out over night, I notice that it seems to taste different the next day.
Is this because dust is falling into the water?
There is an easy experiment to see if your water is picking up dust. Simply cover a glass of water with a saucer and see if it tastes the same as a glass left uncovered.
You don't say where your water is coming from. Is this from a public water supply or from a well?
Either way it is possible that either the water is either absorbing gasses or releasing them.
In years gone by in southern Michigan there was hydrogen sulfide in the water which gave the water a rotten egg smell and taste. The hydrogen sulfide would outgas over time.
Water can be naturally carbonated so that carbon dioxide will outgas.
It is also possible to absorb carbon dioxide from the atmosphere.
This article suggests that your glass of water absorbs carbon dioxide, becoming slightly acidic and changing its taste. There are other such articles on the internets. Other chemicals can breakdown and and cause changes, but since you are using purified water, these are probably not the cause. So....I would hypothesize that carbon dioxide is the culprit here, not dust.
The water tastes differently overnight is not because dust is falling in the water but could rather be that any minuscule amount of dust or other TDS in the water is settling down. If you have ever been to any stream or creek you would have noticed that if you disturb the water the mud at the bottom gets dissolved in the water but if you let the water be still for quite some time it again becomes clear because the dust particles again settle down to the bottom. So this could be happening here and dissolved particles must be settling down to the bottom of the glass thus making water taste different but different taste doesn't necessarily mean bad taste or bad for health water.
Now I'm not sure about your surroundings but if there is no source of dust around and room air is also clean then this could be the plausible reason for water tasting different in the morning.