Well, given that the Earth passed through the tail of Halley's comet in the year [mumble] (too lazy to look it up right now) and there wasn't death raining from the sky, the prediction for similar occurrences is that the effect will be, "nothing much, in fact you wouldn't know it at all if not for seeing it on the news."
Comet tails are impressive, but they are *incredibly* tenuous. They make a fading puff of cigarette smoke look like chunky clam chowder in comparison, density-wise.
The Earth will just plow through it like it wasn't there, which for all practical purposes it almost isn't.