If I am washing out a glass bottle or narrow-mouthed jar, what is the best way to dry it?
Upside down, for the water to trickle out?
Or, right side up, and let evaporation work?
Towels just leave traces of moisture.
Upside down, preferably on a slight angle. It’ll allow the little droplets to converge into bigger drops and flow out, and whatever else will condense at the top(technically the bottom) or sides, pool up more drops and drip out. Ironically if you use a towel first, it takes longer.
For narrow necked bottles, where there is limited air exchange between the interior and exterior, it can sometimes take quite a long time for the last residue of moisture to disappear. If I’m feeling really motivated to make sure they’re dry, I put them in an oven set at the lowest heat setting for a half hour or so.
Hair dryer.
For beer growlers (narrow neck), I do both. I put them upside down in a heavy glass in the fashion that anomalous1 describes. Leave for a day or so. Then I turn them right side up for a day, then seal.
My thanks. 
Dripping out will only work if they’re upside-down. Evaporation will work equally well in any orientation.
You can find some piece of art to hang the bottle on.
A few seconds in a microwave works too.
Why would evaporation not work if bottle isn’t right side up?
Because if it’s upside down (i.e. sitting on the mouth of the jar) it’s basically a closed system.
My experience is that when it’s upside down, even if the mouth is on a grate and exposed to the air it takes longer to fully dry than if it’s right side up or on its side. So I’d say upside down for a few hours, then not upside down after that.
If like the subject line says and you want it to be clean (look clean) dripping out usually will leave less spots. Just my experience though.
This from the OP seems to exclude a closed system:
Maybe Cecil Adams should look into the issue?
There is no clear consensus, & everybody needs to do this, occasionally.
I wash a lot of beer bottles. Of course, after washing, they have to be dried. I have found the most efficient way to dry them is to leave them inverted for 15-20 minutes then stand them with their open tops pointing up. If you only drain them for a minute or so, i can find them with liquid water in them the next day. Water drains much faster than it evaporates, but it will also form a thin film on the glass that seems to resist draining and has to evaporate.
Humid air is actually less dense than dry air (a molecule of H2O is lighter than a molecule of N2), so theoretically, as the remaining water in the bottle evaporates, the air in the bottle would tend to rise. Being inverted, the only way the water vapor to exit is through diffusion in the air. Fortunately, water vapor is miscible in air, so diffusion occurs quickly. However, a long, narrow neck (as on a beer bottle) can slow diffusion, which would slow the drying. In addition, if there is any heating of the bottom (or cooling of the top) of the inverted bottle, so the glass at the top would be cooler than that at the bottom, you can get water vapor to condense at the top, just making more liquid water that needs to evaporate to fully dry. This doesn’t happen as much if the bottle is not inverted.
Maybe not too practicable for home use, but in the lab we would rinse out glassware with warm water, and then squirt some acetone in, swill it round and tip it out. The acetone removes the water droplets, and itself evaporates off very quickly, leaving no residue.