For those who aren't familiar with the book, it is worth noting that Requiem for a God isn't an adventure or module, it's a sourcebook for running a campaign that features the death of a god as an event, just as Forge of War is a sourcebook for running a campaign that involves Eberron's Last War. This doesn't necessarily need to contradict anything about Eberron either. Requiem for a God is very flexible about the effects that a dead god can have on the world and it's worshippers, and offers plenty of options of how to handle the issue.
Let's take the Silver Flame as an example. Let's say that the flames start going out one by one, and begin releasing various evils, and that this is a slow process taking months or years. Now, the thing that I would emphasize in this case is the nature of an individual's faith. At the risk of referencing real-world philosophical concepts, there is a difference in some faiths between worshipping a deity and the concepts they represent, and worshiping an idol, or symbol. In the case of the Silver Flame, where you have a physical manifestation of divine might that anyone can go and see for themselves, I think that there is plenty of room within the church for both types of believers, those who believe in the ideals that the Silver Flame represents philosophically, and those who place their faith in the physical reality of the flame and the power it grants.
If a cleric believes he gets his power from the flame as a physical entity, and the flame goes out, then he no longer has an object or focus for his faith. Without the flame, there is no point, his personal faith is shaken, and he loses his ability to channel divine power. Only by regaining his faith, either transferring it to the eternal ideals of the flame, or switching to another idol or a different religion entirely can he get over his mental block and regain both his faith and his powers.
Other clerics, who believe entirely in the philosophical notions that the flame represents, might see the destruction of the flame as irrelevant to their belief. Even if the last flame gutters and dies, and the world is overrun with demons, they still believe in what the flame represented (Living according to a code of Law and Goodness, and self-sacrifice to defend others) and therefore, they retain their powers even in the face of the "death" of the idol.
A third option exists as well, which is that if you don't know the flame is out, you have no reason for your faith to be shaken. Unless there's some sort of universal "Goodbye cruel world!" message sent out to all the worshipers at the instant the flame goes out, then it is entirely possible for worshipers of a god to not even know what has happened. Of course, as soon as they find out that the flame has gone out, it could still create the crisis of faith as described in the first example. If this is the case, then it may be in the best interests of the officials within the religion to keep the death of the flame secret. If individual faith powers clerical spellcasting, then what the lower ranking clerics don't know can't hurt them, right?
With this in mind, it is entirely possible that as the flame dies, some archbishop or whatever loses his powers and is devoured by a demon and the church is razed by high level monsters, while in other temples, "True Believers" manage to retain their powers and fight back against the forces of darkness. In my opinion, since Eberron is a world with very few high level characters, it doesn't seem that the fiends held in check by the silver flames are unbeatable epic creatures. I mean, Coatls and Rakshasas aren't exactly the epic powerhouses of D&D, and there's only so much you can do with HD advancement and class levels. It seems likely to me that the prophecies of doom and the difficulties faced in the Age of Demons result more from the lack of high level clerics, prestige classes, spells, and feats which have since been developed to combat demons. There's only so much a bog-standard MM1 dragon can do against a Rakshasa with all their immunities to magic, and Great Wyrms should be considerably rarer in an earlier time period. At the risk of sounding like the tagline for Cthulhu Tech, the prophecies of doom were written when they had sticks and stones, now we have nuclear weapons and railguns.
The Death of the Silver Flame could also embolden Thrane's old enemies, if they find out about it. During the Last War, Aundair in particular suffered from the excesses of the Thranish Inquisition, justifying atrocities in the name of divine right. With the nation reeling, and their faith shaken, it is the perfect time for Aundair to launch an assault. Even if Aundair's leaders are aware of the escaping demonic forces, they might trust in the powers of their wizards to contain the demons once the nation has been secured. Of course, once the truce is broken and Aundair invades Thrane, then it is only a matter of time before Breland and Karnath are drawn in to a next war.
Granted, this all seems very apocalyptic, but it is entirely possible for a group of resourceful PCs to sort it all out by battling demonic forces, finding ancient artifacts (perhaps even the blood and flesh of a god's avatar slain in the prehistory of the world) that might bolster the remaining flames or capture the unleashed demons, and engaging in political intrigues to prevent other nations from acting on Thrane's relative weakness during this period. The PCs might even be able to solve this problem by enacting some epic ritual. Perhaps a group of high level True Believers in the Flame can rekindle it's power by sacrificing themselves to fuel it, just as the coatls did before? While the clerics perform the ritual of sacrifice, the PCs might have to guard them against a horde of demon servants bent on ensuring that the ancient masters are freed. Or perhaps the PCs might make this sacrifice themselves, giving their lives to ensure the safety of Eberron. The opportunities for epic heroism are there, as well as the threat of apocalyptic doom.
Robert "Obviously, I vote for the Silver Flame, but the Undying Court is my second choice" Ranting