There are a few principles involved here (a chemical engineer's perspective on leaching - essentially coffee brewing is just that).
As Eli mentioned, solubility in water for most materials increases with temperature. In this case of leaving a batch of coffee with a batch of water (assume pure water for simplicity), materials (different components) will diffuse out of the coffee driven by concentration differences for each materials in water and in the ground coffee. There is a constant exchange of these materials going from the ground coffee into water and going from what has gone into water back to the ground coffee. Each material has its own diffusion rate and solubility in water for a given temperature. Eventually, there will be an equilibrium. Thus, assuming no chemical reaction takes place for now, no matter how long you leave that brew, temperature alone will dictate how much of each material in the coffee stays in the water and how much stays in the ground coffee. Without facts, qualitatively, I imagine an overnight brew in the fridge will give you more than enough time to reach that equilibrium and brewing it for another day or two will not change anything.
However, many organic components will hydrolyse to some degree when exposed to water especially when there is a pH shift. I do not know enough to speculate on which component in the coffee will face what kind of hydrolysis at what temperature, but generally the hydrolysis reaction rate increases with temperature. I am speculating that undesirable flavours are about both products of hydrolysis and increased solubility of "bad" components. There is also the likelihood of other reactions at higher temperatures say between acids and lipids or proteins and sugars or perhaps the breakdown of polysaccharides into shorter sugars and oxidation of sugars to acids and so on. These are much more likely to produce flavours than denaturing of proteins which I think (not certain) tends to have little to no impact.
In order to systematically visualise this, I think one would go through a matrix of temperature, time and grind size and put the "extract" through an HPLC to quantify the concentration of each component. Ultimately, a sensitive nose/tongue is the real test.
Extrapolating from this, I always feel that the drip method has to be the least desirable as you are constantly putting pure water to leach out materials from the coffee, always encouraging more materials to come out leaving no room for "bad" stuff to be left behind. If you taste the drip as the brewing progresses, you can definitely tell that about half way through, the coffee actually tastes foul and it gets worse. This is a leaching process designed to maximise extraction. This is exactly what happens at the fresh solvent end of an industrial counter-current extraction process to maximise concentration gradient and diffusion rate - not what you want for just taking the desirable bits out of coffee. It would be preferable to recirculate the brew through the grounds so that "strength" builds up to an equilibrium and by adjusting the temperature, you can control that equilibrium to achieve "optimal" taste.
I have been experimenting with cold brew and find that the brew tends to lack aroma while hot brew tends to have both a harsher taste and distorted aroma. Ideally, if there is a way to preserve the aroma of freshly roasted beans, that would be best. So far, I favour as fine a grind as practical, start with room temperature water, agitate and gradually raise the temperature to above 60C (have not found the preferred temperature yet but I dare not push it over 90C) nearer to the time of drinking. Once filtered, I can heat or chill the coffee as desired.
(Editor's note: Additionally, an anonymous user suggested the following content...)
Cold/warm brew is much more forgiving, relying on achieving some equilibrium and making grind size and time redundant variables. Hot brew on the other hand for everyday coffee making is mostly about guessing when to stop the leaching process abruptly.
Fine grind is sometimes falsely blamed for over-extraction bitterness. From my experience, bitterness from fine grind has mostly to do with small coffee solids staying suspended in the coffee. It is very telling if you make coffee using a fine grind (say espresso grind) in a press/cafetier, pour a third into a cup as control, another third through a filter, and double filter the remainder, you will be able to taste the difference.
