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The video ignited a furor over Trump's treatment of women, some of whom have come forward to accuse the Republican presidential candidate of unwanted sexual advances.
Bush, a nephew of former president George HW Bush, was a host of celebrity show "Access Hollywood" in 2005 when he and Trump were taped in conversation as they arrived at the set of a soap opera.
Bush had joined NBC's morning flagship "Today" in August.
"Bush, a host on NBC's 'Today,' is leaving the show effective yesterday, according to an internal memo sent to staff by Senior Vice President Noah Oppenheim, who oversees the broadcast," NBC News said.
"I am deeply grateful for the conversations I've had with my daughters, and for all of the support from family, friends and colleagues. I look forward to what lies ahead," Bush said in a statement.
On the video, Trump is heard bragging that he uses his fame to grope and kiss women without their consent, as Bush laughs and jokes with the real estate billionaire who became a reality TV star with "The Apprentice."
Bush, who has said he is "embarrassed and ashamed" of his role in the incident, was making $3.5 million a year in his current position at NBC - hosting an hour of morning news heavy on entertainment content.
Trump has apologised for his remarks, but insists it was just "locker room talk."