There are two main discussions in the gemara about setting up before Shabbat situations in which a process of a melacha will occur on Shabbat without further human involvement.
One passage[1] discusses leaving food on the fire to cook (shehiya). Specifically, the gemara deals with the question of when it is required to take steps to deal with the concern that someone will raise the fire on Shabbat. One opinion maintains that if the food has reached the state of k’ma’achal ben d’rusai (nominally cooked), it may be left on the fire normally; another opinion requires removing the fuel source or covering it. Apparently, if the fire is covered so that there is no concern of stoking the coals (or the equivalent), one could leave food on it at any stage of cooking. According to a wide spectrum of poskim,[2] a non-adjustable heat source (e.g., a “Shabbat hot plate”) requires no covering even when the food it is cooking has not reached k’ma’achal ben d’rusai. From this perspective, if a coffee maker has only one level of heat / speed of brewing, even if it has many other settings and controls, setting the machine to activate the brewing process on Shabbat would be permitted.
Another gemara[3] deals more broadly with systems set up before Shabbat that would be forbidden if set up on Shabbat. The gemara says that if one places wool in a cauldron of dye before Shabbat, he must seal the lid before Shabbat due to a concern that he will stir the cauldron. Regarding uncooked food left on the fire from before Shabbat, the gemara refers only to a problem of stoking coals and not of stirring. Rabbi Akiva Eiger[4] suggests that if the food started cooking but did not reach k’ma’achal ben d’rusai before Shabbat started, stirring would indeed be a concern (which a blech or a non-adjustable heat source would not obviate). Although the Bi’ur Halacha[5] mentions this stringency, it seems that the vast majority of present-day poskim accept the Shulchan Aruch’s ruling[6] that when the Rabbinic concern about stoking coals is taken care of, even uncooked food can be left on the flame.[7]
Nevertheless, the Tzitz Eliezer[8] forbids putting uncooked food before Shabbat in a place where a heat source will be activated by a timer on Shabbat. One of the bases for his ruling is the Ramban,[9] who deals with the question of why the concern of stirring is not raised regarding cooking food. One of his answers is that the Rabbis were concerned about stirring only in regard to dye and not in regard to food, even though it is forbidden to stir uncooked food on the flame. However, according to the Ramban’s other answers, it is forbidden to allow a situation in which the cooking of food starts on Shabbat and one might come to hasten it through stirring. Rav S.Z. Auerbach[10] responded that we accept the Ramban’s lenient answer – that we are not concerned about stirring – allowing the possibility of timers that start the cooking process on Shabbat. We may rely on this opinion.
Rav Auerbach would certainly permit use of a coffee maker through a timer, as you described. Regarding machines in which it is not feasible to stir the coffee as it brews, all should agree that this is not grounds for a prohibition.
A final issue is that the Rama[11] forbids operating from before Shabbat a mechanism that is forbidden to operate on Shabbat if it makes noise, as this is degrading for Shabbat (avsha milta). It is permitted to set up such a mechanism only if people often do so in advance and there is thus no reason to suspect that Shabbat desecration occurred.[12] Since coffee makers are usually not operated on a time delay, this could pose a problem. However, most machines are probably not loud enough to cause a prohibition, which would pertain when the noise can be heard in another room.[13]
There are (and will be) many models of coffee makers, so one must ensure that the one he uses meets all the requirements. Do not assume or quote us as giving a blanket leniency in all cases.