He told a BOA board meeting he would leave at the end of this month to pursue other opportunities.
Hunt, who helped manage Team GB as chef de mission at the London 2012 Olympics, said: "I will always be proud to have led Team GB at a home Games and of the fantastic performance of the team this summer and the way in which the BOA supported the athletes.
"Now that the Games are over, and the organisation's strategy for the next four years is taking shape, the BOA's responsibilities and the job of the CEO are different, and I feel it is the right time for me to move on."
New BOA chairman Lord Coe said: "Team GB had the very best facilities, care and preparation at the London 2012 Games thanks to the tireless efforts and passion of Andy Hunt and his team over the last four years."