In Guns, Germs, and Steel, what is "the dilemma fundamental to all centrally governed, nonegalitarian societies?"
The dilemma that Diamond is talking about in the part of Chapter 14 that you cite is the dilemma of how elites can manage to take from the common people without losing the support of those people.
Diamond says that all centrally governed, nonegalitarian societies have to have governments that essentially take resources from the common people. These are governments that are run by elites and are taking from the commoners. Diamond argues that all governments do this and that the ones that we think of as "good" are ones that don't take too much away from the commoners and which do some popular things with the resources that they do take.
The dilemma, then, is faced by the governments. They have to take money away from the common people without making those people angry enough to rise up and throw them out. Diamond says that they can do this in any of four ways that he lists on page 277 in the paperback edition. However, they will all still face this dilemma because it is inescapable for any society larger than a tribe.
Nicolo Machiavelli said it something like this.
A prince, wishing to maintain among men a reputation for generostity, is obliged to avoid any attribute of magnificence; so that a prince thus inclined will consume in his generocity all his property, and will be compelled in the end, if he wishes to maintain his reputation for generocity, to unduly weigh down his people, and tax them, and do everything he can to get money. This will soon make him odious to his subjects, and becoming poor he will be little valued by any one; thus, with his generocity, having offended many and rewarded few, he is affected by the very first trouble and imperilled by whatever may be the first danger; recognizing this himself, and wishing to draw back from it, he runs at once into the reproach of being miserly.