They can own it, actually. If you use the characters of Bugs Bunny, etc., or the setting (do they have a canonical setting?) then Warner does own the rights to the material you’re using.

For example, see how the original Winnie the Pooh material just entered public domain, but the subsequent Disney versions have not. You can use the original stuff (see the recent horror movie for an example of legal use) but not the later material like Tigger or Pooh in a red shirt.

Now if your work is satire or parody, then you can argue that it’s fair use. But generally, most companies don’t care about fan fiction because it doesn’t compete with their sales. If you publish your Harry Potter fan fiction on Livejournal, it wouldn’t be worth the money to pay the lawyers to take it down. But if you publish your Larry Cotter and the Wizard’s Rock story on Amazon, they’ll take it down because now it’s a competing product.