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Probably won't work because the application isn't just installed under the C:\Program Files\WorldofWarcraft but there are usually several other registry changes which exist on the C:\Windows system folder.
Even if the game did have that issue, which it doesn't, it wouldn't matter. I've installed several games in various drives within my system and never had a problem. As far as the computer is concerned, a hard drive is a hard drive. If it needs to make any registry entries, it'll do that regardless of which drive it's installed.
With WOW, just like what others say, you can just copy the folders to a new drive and play away so the external drive will be fine...
I agree with what the others have said too, if you're going to go that way, USB 2.0 or IEEE 1394 to ensure the transfer rates are there and don't impact your game.