As it's already been mentioned, there are plenty of demons in Lord of the Rings. Balrog actually means "power demon."
Also keep in mind, Gandalf uses occult magic. Despite the fact that he is a Maia, closest analogy to Christianity being a lesser angel, he and the other Istari were forbidden to use their powers as Maia, and instead were confined to the limits of humanity (as such they must eat, drink, sleep, etc) plus they can succumb to the moral limitations of men, as seen in Saruman's greed for power and Gandalf's short temper.
That means that all the spells Gandalf uses are magic just like in any other D&D style fantasy. As he says at the gates of Moria when he can't remember the password, "I once knew every spell in all the tongues of Elves or Men or Orcs, that was ever used for such a purpose."
So, if you have a conviction that Warcraft 3 is 'demonic,' I don't see why you shouldn't feel the same way about LotR. If it weren't for the fact that Tolkien was a devout Catholic, I imagine LotR would be denounced on every Christian review website, but I guess if a Christian makes something 'demonic' it's ok.
Hey, you could always do some background research on the staff at Blizzard - maybe you will find some obscure graphic artist or co-producer who worked on WC3 who claims to be a Christian, and then it will be ok, right?