Why did George HW Bush fire Cheney?

Answer 1

He didn't. And he can't because the Vice President is not an employee of the President.

Although Dick Cheney was Secretary of Defense for George HW Bush, incoming President Bill Clinton selected a replacement for Dick Cheney.

Answer 2

George H.W. Bush did not fire Dick Cheney. In fact, he appointed Cheney to be his Secretary of Defense from 1989 to 1993. However, Bush did not choose Cheney to be his running mate for his re-election campaign in 1992, instead selecting Dan Quayle.