And to make potable water, actually. Would this work in-paradigm? I was under the impression that infections were caused by demons getting into your body, and that (unboiled) water was of the same.
Or would more demons just come in immediately after the casting, and get into the body in the same manner?
Some diseases are caused by imbalances in the humours rather than by spirits or demons. Theoretically, some diseases must be caused by infernal spirits so you could use infernal-blasting spells to affect them (after all, that's how medicinal elementalists cure disease), but if I remember rightly DeO has difficulties with spirits inside people's bodies. That's why infernal plots suggest using demons possessing people to get inside Aegis of the Hearth. I'm sure there have been several topics on here arguing exactly when and how you can use Demon's Eternal Oblivion to attack spirits inside people.
Unclean water can have things in it that will cause harm other than demons.
And killing disease demons won't necessarily cure the disease they've caused. It might well help, as they'll no longer be working against you, but the damage has already been done.
Hm. The rules for demonic possession are on RoP:I, pg. 32-33. The only comment I can find about vim spells is right at the end of the section (pg. 33, left-hand column), which says that you can use vim spells to target the possession - but that it affects the Possession pool, rather than the Magic Might pool. But as once that pool reaches 0 the possession ends, so the end result is the same thing: no more possession. I suppose you may want to blast the demon a couple of times to ensure that it says dead (and not just drained of magic points), but other than that it seems like it would work.
Ah, good point. And in actually cracking open the A&A book and looking at the disease section (pg. 45), it seems that this is explicitly laid out there.
In general you can always DEO first and ask questions later.
Fixed that for you. 
This worked in a saga I ran a couple years ago. A Byzantine general was ill with a demon-caused disease. The players managed to wear down its pool and kick it out, then used traditional CrCo spells to cure the lingering effects of the disease.
Sometimes, yes.
Sometimes illnesses/diseases are caused by demons (who directly act to inbalance the humors) othertimes a character's humors get out of balance via "mundane" sources (too much standing around in the rain, for example). So, on a case-by-case basis it will depend on whether some water (say) was bad because a demon was present and causing it to be bad, or whether it was bad for a "natural" reason. Also, as others have said, eliminating the demon might not make the water good, it just won't get any worse once the demon has gone.