It depends on the game I'm running, really. Gods don't have a set place in most of them.
Generally, though, gods can't be killed in my games, outside of completely extinguishing whatever concept they get worshipped for. You could no more kill the God of Bravery and Longswords than you could kill bravery and longswords. You'd have to completely rid the world of bravery and completely and forever eliminate all longswords.
My general take is that a god is at least as enduring as whatever concept it represents. They tend to be creator figures who made the concept to begin with, so are at least as potent as it is. Some gods might conceivably be murdered; a god of squirrels is conceivably killable, as squirrels can be tangibly genocided. But a god of knowledge is a trickier subject, as it entails destroying all books and the like, as well as all sentient species, and even then...the god might endure, as the concept of knowledge is a little more abstract than squirrels.
However, sometimes I might go with a more Neil Gaiman-esque take on things. So that the concept endures, but its representative can be killed - but will be replaced in a fairly swift amount of time. And there'd be little point in anyone trying for the position, as it would simply pick the most appropriate representative, anyway, or someone potentially groomed by the now deceased god for that position. So if the god of Dreams died, bam, his right hand rises up, or some other being already greatly in-tune with the concept of dreams is empowered.
In this case, there still would be a shake-up, though; the new representative won't necessarily be exactly the same as its predecessor. In fact, it probably won't be. Worshippers will be thrown into turmoil as a new set of ethics and morality rises up (conceivably in opposition to the previous ones). Any alliances between other gods and enmities as well will be up in the air. Similarly, any effects related to the deities domain might change - if a rather kindly deity of weather dies and gets replaced by a more tempestuous ones, generally mild weather would take a turn for the worse. Even if the two gods share some similarity, the period of transition's likely to be tumultous as the new deity gets used to the position, with its domain of power reflecting the confusion and awkwardness - odd weather might occur for a while, or particularly odd dreams might be prevalent, or whatever.
Faith probably wouldn't crumble, though. There's no such thing as a priest in my game who worships a deity for the power alone. A cleric who would switch faith just because the god dies and no longer grants spells would never receive those spells in the first place; the clerics receive the spells because of their faith, not vice versa. So the priests would keep preaching and, in all likelihood, the laymen would keep listening. The religion around the god might eventually disappear, but it would be a slow process.
Anyway, I might elaborate a bit more on what I'd do with other forms of gods, those that don't fall under the types I've mentioned here, but I'm going to cut out for now.
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