Adams and Portsmouth have suffered a run of nine games without a win and the club is struggling just a point above the relegation zone.
“This has been a very difficult decision and Tony has worked tirelessly to arrest the slump in form,” a club spokesman told the club’s official website.
“He is rightfully highly-respected within the football world and played a major role in our FA Cup triumph last season. We would like to thank him and John and wish them well in the future.
“The owner and the board feel they have been as supportive as they can during a period where results have been poor. The team has played well but too many points have been dropped from winning positions.
“Notwithstanding the adverse economic conditions the board remains committed to the continued growth and development of the club for the future, building on its achievements of the last three years.
“The priority is for Portsmouth football club to remain in the top flight and we feel the appointment of a new manager will give us the best opportunity to enable us to do this.
“Paul Hart knows the football club, having joined us nearly two years ago, and is a respected coach with managerial experience.”