If boiling and other heat methods cause a deterioration of flavor, What is the fastest way of accelerating the evaporation rate of alcohol in a liquid like beer?
You could try ice distillation: Stick it in the freezer, and skim off the ice. Keep the ice, discard the liquid and melt the ice.
If you had a freezer that you could precisely control the temperature, I believe there are certain preferred target temperatures, but simply checking on it frequently in a standard freezer should do the trick.
The vacuum based suggestions also sound good, if you have appropriate equipment for that.
Basically, you're asking about distillation in reverse. Traditional distillation is primarily boiling, but there's also a few traditions (applejack, eisbock) involving ice distillation. In ice distillation they're after a higher alcohol product and keeping the liquid (discarding the ice), but I believe if you switch which part you keep you'll get the same results.
Much like any other distillation, you may need to run through the process multiple times to get the desired result.
With beer, the primary flavoring is from compounds extracted from hops, which react with oxygen to produce unpleasant flavors. By "unpleasant" I mean turning the bitter and tangy flavors into something more like wet cardboard. So depending on what you're doing you may also need to protect your beer from exposure to oxygen. I believe with beer the real enemy is oxygen, with heat or light accelerating the undesirable reactions.
Depending on the style of beer, esters and phenols may also be important to the flavor. In styles where those flavors are desired, it's typically a wide mix, and it's quite possible some of them will evaporate easier than alcohol or be harder to freeze.
Note that no distillation method is "perfect", with all distillation methods you'll have water and alcohol in both outputs of the process, you'll just have a greater amount of alcohol in the evaporated or unfrozen part and a lesser amount in the unevaporated or frozen part.
Note also that in the US any process for separating alcohol from something else is illegal at home.
The only thing I can think of: lower the air pressure in the container it's in. Probably not practical.
Let it sit in an open container and let it evaporate. Of course, the water in the liquid will also evaporate. Heat speeds this process.
Evaporation would work in theory, but unless you have expensive vacuum equipment, it isn't practical. I would look to using non-alcoholic beer, rather than try to remove alcohol by a means other than heating.