The 40-year-old took charge as interim boss after Scott Parker was sacked in August, then was appointed on a permanent basis in November.
O’Neil won 10 of his 34 Premier League games in charge last season, guiding Bournemouth to a 15th-placed finish.
“This has been a difficult decision,” Bournemouth owner Bill Foley said.
“It has been made with great consideration to best position ourselves ahead of the coming season.
“Gary will go on to have a long career as a head coach or manager, but we feel that, at this moment in time, a change is in the best interests of this football club.”
Bournemouth have said they will announce O’Neil’s successor “imminently”.
O’Neill took the helm at Vitality Stadium after Parker was sacked following a 9-0 hammering by Liverpool – the joint biggest defeat in Premier League history.
It was the club’s third successive defeat, but O’Neil went unbeaten in his first six games in charge.