It was the latest turn since reports surfaced in March that the Justice Department was conducting a probe into Burr's sale of up to $1.72 million in stocks earlier this year.
The stocks were sold in early February after senators received closed-door briefings on the national threat posed by the coronavirus, before most Americans were warned about the potential economic fallout of the pandemic.
Burr has denied that he used any information he learned in his capacity as a senator to guide his decisions regarding selling his stocks.
But the questions around the sales escalated on Wednesday night after the Los Angeles Times reported that the FBI had carried out a warrant to confiscate Burr's cellphone as part of its investigation.
Burr and McConnell both said that the North Carolina senator called the GOP leader on Thursday morning to inform McConnell that he had decided to temporarily give up his chairmanship — a decision that McConnell said he agreed with.
Burr also addressed the decision during a closed-door GOP lunch on Thursday, telling his colleagues that he decided to step aside "so that this didn’t cause a difficulty on the committee or within the conference."
Republicans largely held back from questioning Burr on Thursday, with several saying that he deserved a presumption of innocence pending the outcome of the FBI investigation.