Context: I am planning to apply to a US lab.
Many universities have mandatory requirements for letters of recommendation (LoR) for a postdoc position, and that's the main reason they ask you to submit LoR. There were some PI who interviewed me told me this and also told me that they wouldn't even care what my referees would write about me. They were willing to hire me immediately after they have three letters of recommendation. If you have solid credentials as a researcher, I think your LoR would not matter much. However, a terrible remark on a LoR would also significantly influence the hiring decision.
This question is too broad to answer. Many PIs will not hire a postdoc without strong letters of reference. On the other hand some PIs will not care. Although I suspect in this case you would have to be an obviously very strong researcher. (Even though personally I find it hard to believe that references wouldn't be asked for. Even if you're a star researcher maybe you once assaulted a fellow student and tried to burn the lab down? That's something one would typically find out from Letters of reference.)