Generally, when Blizzard goes after private server owners, they go after them for copyright and trademark infringement. In fact, the very popular case of MDY Industries linked above is exactly that: trademark infringement, copyright infringement etc.

The silent rule is "don't tarnish the Blizzard trademarks". It's generally the most unethical servers with tons of microtransactions that get taken down first. This is also why plenty of them are left alone.