While I was taking a class I had an extractor spring split in half and it did not extract the casing when fired. The bolt was disassembled and cleaned (the spring was inspected) before the class began, so I knew it wasnt broken at the time. This bolt did not have an o ring. A quick replacement of the spring and I was back in business. So in my experience I will say no an AR wont run with a broken extractor spring.
edit: To add, I had the FTE at about the 10 round count. This particular rifle had about 5-6K thru it.
