Is a private server for World of Warcraft illegal

At least in the United States, and in parts of Europe, this answer is definitely yes. A private server exists because of reverse engineering, which is clearly against the terms of use and end user license agreement; any country that legally allows software vendors to place a "you must not reverse engineer our software" clause in an EULA (End User License Agreement) or TOU (Terms of Use), and supports the concept that a EULA or TOU is a legally binding contract, will likely rule in favor of Blizzard on the question of legality. In any case, should Blizzard find out you have one running, you will most likely end up in court.

However, with a few exceptions, private servers are generally not "stealing", and may actually be bolstering Blizzard's profitability. Those servers that do not charge a monthly fee and do not provide a copy of the client may have an arguable legal defense in court. They are not profiting from Blizzard in any way, and most players that play on private servers still hold active WoW accounts. A server operating under these conditions would have a total financial liability of zero in regards to theft of services; it can be argued that there is literally no revenue loss, thus no actual theft occurred.

However, that still leads to theft of Intellectual Property. Even though the server is free, and open source, and contains no licensed code, it still must communicate over an internally documented protocol, which is definitely IP. Reverse engineering protocols that are not officially publicly documented is IP theft, and subject to laws regarding IP. Not all countries recognize this, but most do, so there is still some illegal activity going on. Blizzard has a legal obligation to attempt to pursue all infractions, because if they fail to defend their IP, it may become an abandoned IP, meaning they would forever lose the ability to legally protect their service in court.

Overall, you should consider a private server for WoW as illegal, because even though there may be moral justifications for it (i.e. learning how to write your own server for a game you want to develop, want to play when maintenance is going on, want to play on a server that doesn't have 500ms+ lag, etc), the bottom line is that in most countries, you can be dragged into court for playing on a private server or operating a private server.