Yes, but its much easier to shut down your website than find random internet guy posting through a VPN. Like Twicehorn said, if you accept random people, you run the risk of being abused. If you decide to restrict access, you've already reduced content. Are you going to require SSN, DLs, some other kind of identifier that someone smart can still work around? The website or app is a fixed target. Posters will always be harder to nail down, so the former will always bear the brunt of any legal action without 230 protections.
Ant is dead on with the comparison to DCMA abuse. The operator will bear the brunt of any legal actions. Step one will be serving the site with a violation notice, then requesting the site turn over the offender's information. Cant find johnny internets because he fed you a fake SSN, DL, whatever? Too bad, I guess Surly gets to eat the legal problems.
Repealing 230 will have people screaming that leopards ate their face once they realize the full impacts of that decision.