Trump will stay in office and likely finish out his term because it takes a Senate conviction to remove him, even after he’s been impeached.
This would leave open the possibility that Trump could claim that a vote to convict him on the impeachment was invalid because his term of office expired rendering the proceeding moot, although there is apparently one historical precedent of someone who is impeached while in office whose term expires before the conviction proceeding in the Senate is completed being convicted on the impeachment.
Once President Biden is sworn in on January 20, 2021, Mitch McConnell will become the Minority leader in the U.S. Senate, since Vice President Harris, a Democrat, will hold the tie breaking vote, rather than Vice President Pence, who holds that vote now. So until then he has the power to delay an impeachment trial in the Senate. The Senate is currently evenly split with 50 Democrats and 50 Republicans.
Vice President Pence has declined in writing in response to a request from the House of Representatives in a resolution to remove Trump under the 25th Amendment, for Presidential disability, and Trump has not indicated a willingness to resign from office before the end of his term, so far.