Why does coffee taste awful after reheating it in a microwave oven?

It has nothing to do with the microwave and everything to do with the volatility of aromas and flavors in coffee. Even coffee kept warm for 4 hours won't taste very good. In my experience, stale brewed coffee results in a more pronounced acidity, if left out to cool, or a woody, muddy, bitter kind of flavor, sometimes with more pronounced acidity, if held warm. If you were to make a straight espresso, it's optimal to drink it within 20-30 seconds. Brewed coffee can usually be held warm around 30 minutes before pronounced deterioration becomes obvious.

A popular Japanese TV show suggested perhaps stirring a pinch of salt in to stale, reheated coffee, which may trick your tastebuds enough into tolerating it. I've found that tactic less than adequate, but your mileage may vary.

There's a ton of information on coffee flavor compounds and other coffee-related research at: http://www.coffeeresearch.org/science/aromamain.htm

In addition to Jason's excellent answer, I'll add that if you're leaving coffee out for 4-6 hours before reheating it (from morning to afternoon) then the oils in the coffee have probably gone rancid as well. Coffee oils are highly volatile and require only a few hours to go rancid. This is why press-pot coffee often tastes vile no matter what quality beans you use; the coffee pump is full or rancid oils.

Strangely, I ONLY like re-heated coffee. The older the better. My ideal cup is one to three day old Dunkin Donuts black coffee reheated in a pot on the stove or the microwave. Deliciously nutty without he odd bitterness in fresh made coffee. Then I add milk or cream.--I know, I'm alone on this one.

Coffee's flavor consists primarily of highly volatile substances. Human taste largely depends of smell; when you waft some coffee aromas, these volatile substances are entering your nasal cavities and being sensed. Even while drinking, trace aromas are floating up into your nose. When left out, these chemicals evaporate, and disperse into undetectable quantities within the air. After reheating, even more of these aromas are removed, meaning that the coffee you drink is devoid of many of its flavor components. Once these are gone, the bitter taste becomes dominant, and it makes the coffee taste very unpleasant.

Well, I think it really is because the coffee extract (espresso) burns if direct heat is applied. If you were to reheat your coffee on a stove in a water bath, the taste may not be so different.

This happens even with freshly made espresso - if the milk isn't hot enough and you want to reheat your coffee, you'd screw up the taste big time.

Another thing is that Arabica is more prone to the burnt taste than Robusta strains (the round beans).

Most coffee shop websites say not to reheat coffee. It is because when you brew the coffee, the water temperature is 205F.

I always reheat my coffee for about 10sec -25sec.
for a cup - heavy duty 20-25sec -light 10-15sec
for a mug- 10-20sec.

I might say that it's perhaps your choice of making coffee in a percolator. A percolator takes boiled water and runs it through coffee grounds. Then it recycles that water and now coffee mixture back through the same coffee grounds. With each successive pass of the liquid it concentrates the coffee/water mixture.

This also changes the coffee grounds as it makes them more and more bitter from the acids that are being released with every successive cycle. Further more the liquid is continually being heated and kept at the boiling point, making for more coffee acid extraction.

This particular choice for coffee making, produces a strong acid component and darker more concentrated flavour. I think it is the very nature of making percolated coffee leads to it being particularly bad tasting when cold. The concentration of oils, acids and other bitter compounds are more noticeable when cold than hot. Also the heat probably makes you take less of it in your mouth, than when it's cold.

Usually when I reheat coffee in a microwave at the lowest power setting (just one notch above the defrost setting) it's drinkable. At higher settings the taste is completely destroyed.

No question about it - heat destroys coffee. I'm amazed that fast food places and gas stations are able to sell so much boiling hot coffee. It's so bitter and has that rubber aftertaste - hardly resembles real coffee IMHO.