If someone starts posting threats or illegal material as discussion comments, wouldn't Hacker News be responsible for deleting it or else they would face penalties? Uber argues they shouldn't face penalties when an Uber driver does something illegal since they are "independent contractors".
Not really, under U.S. law HN or any other web forum is only responsible for deleting libelous or infringing comments after being notified: Digital Harbor, DCMA, etc.
The analogous situation for Uber would be "allowing a driver to continue using the platform after being notified of illegal on-the-job activity."