In the legal world, whenever text is set in fixed form, it is automatically copyrighted by its author, whether they claim it or not, unless they specifically say otherwise. "Specifically say otherwise" probably includes anything they might have agreed to in the TOS for a particular site, although the legal waters there are largely untested. The only exemption to this auto-copyright is statistics, phone numbers, or other non-copyrightable content (see CBC vs. MLB).

IANAL, but depending on the nature of your service and specifically how the content is collected, you may qualify for DMCA safe harbor protections. This means that if you remove things in a timely manner upon request nobody can sue you. This is how Google caches the whole internet without getting sued.

That's all legal mumbo-jumbo. The real world answer is that some people will get mad regardless of the law, so take their content down and apologize. Follow robots.txt guidelines. Don't post takedown replys a la PirateBay. Generally act sane. If you do all of the above you'll probably be ok.