Wouldn't the worst case be the underlying IP address changes, and your site now points to some other service? Since Heroku doesn't offer static IPs, I don't think.
Heroku seems to use the Host header on the HTTP request to determine what customer the request is sent to, so I don't think it can end up somewhere it shouldn't be unless the IP leaves control of Heroku entirely and is picked up by another AWS customer.
Heroku uses the host name to route requests. So you wouldn't serve another site, it would just give you an error that your site wasn't found or something.