Mr. Williams will remain at Fox News as a senior political analyst; the network said in a statement that he had requested to be closer to his family in Washington rather than commute to New York, where “The Five” is taped. Fox News said another liberal host would replace him. Among those in contention is a newly hired contributor to the Fox stable, the former Democratic congressman Harold Ford Jr. Mr. Williams departed after a harder edge had crept into his exchanges with colleagues like Mr. Gutfeld and Jesse Watters. “The Five” had long been a venue for heated, if friendly debate, but Mr. Williams was repeatedly mocked and shouted down when he accused Mr. Trump of lying about the election and fueling the riot at the Capitol on Jan. 6. Mr. Williams also noted, on-air, a Fox News report about Mr. Biden that falsely claimed he wanted to restrict Americans’ consumption of hamburgers. (Fox News later issued a correction.)