Which CAPTCHA method are you using?
As troester says, the basic method works by directly running a script out of the plugin folder to generate the image, rather than going through the usual /index.php?option=com_fabrik&view=something..., and as such it may be getting blocked by either an htaccess file, or a 3rd party J! security module, or a firewall setting, which restricts scripts from being run from anywhere except the main front and backend index.php.
If that's the case, you'll either need to whitelist that plugin folder, or use one of the alternate CAPTCHA methods, like recaptcha, nocaptcha (recently added in github) or playthru.
-- hugh