O'neil guided Bournemouth to Premier League safety after replacing Scott Parker in August last.
The former Portsmouth and Middlesbrough midfielder won 10 of his 34 league games in charge of Bournemouth as the Cherries finished in 15th position.
O'neil initially took charge of Bournemouth on a temporary basis following a 9-0 defeat to Liverpool in August when Parker was still at the helm and was handed the reigns permanently in November on an 18-month contract.
Bournemouth owner Bill Foley said in a club statement: “Gary’s achievement last season is one I will always be grateful for.
“This has been a difficult decision, but it has been made with great consideration to best position ourselves ahead of the coming season.”
“As a club, we have put plans in place for long-term success with improvements being made to infrastructure, most notably the development of a new state-of-the-art training facility and the ongoing discussions around upgrades to our stadium,” continued the statement from American Foley, who completed his takeover of the south-coast club in December.
“We have also identified a number of significant targets in the transfer market this summer and believe this change in direction will provide us with the best platform from which to build.
“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.”
Spaniard Iraola has signed a two-year contract at the Vitality Stadium after leaving Rayo Vallecano.
The 40-year-old was linked with Leeds United earlier this year when a replacement for the sacked Jesse Marsch was being sought at Elland Road.
Iraola guided Rayo Vallecano to an 11th place finish in La Liga last season.
He leaves Rayo after three seasons at the club.
Cherries chairman Bill Foley said: “We’re so excited to welcome Andoni to the club. With his contract in Spain coming to an end this summer, we wanted to act quickly.
“He was highly sought after by other clubs across the continent, and his style of play has been an important factor in making this decision.”
“His achievements in Spain have certainly been very impressive, and we’re confident that he is the right man to lead our next chapter,” continued Foley.
“We have identified a number of targets in the transfer market, which along with our January additions will supplement our strong playing squad to give Andoni the best chance to consolidate our Premier League status and keep progressing.”
