The quartet has not been offered new deals with the Power, with Bonner leaving Alberton after eight years with the club.
The defender would play 93 games for the Power across his career, including 11 games in 2023.
Dumont's time with Port Adelaide comes to a close after two years with the Power, having joined the club ahead of the 2022 season after leaving North Melbourne.
Dumont would play eight games for the Power, adding to his 113 with the Roos.
Barkla would not feature at AFL level in his 12 months with the club, which he joined as a graduate of their Next Generation Academy.
Teakle was drafted to the Power midway through last year as a Mid-Season Draft acquisition and would play six games over 18 months in the AFL, with injuries preventing further minutes in the top flight.
“It is difficult to farewell players of such great character who have dedicated themselves to our club,” Power list manager Jason Cripps said in a club statement.
“We appreciated the hard work and commitment that Riley, Trent, Brynn and Nathan put in across multiple years with us.
“We wish them every success with their next steps and thank them for everything they have contributed to our club.”
The four departures follow the retirement of captain Tom Jonas, with the defender bowing out following Port's semi-final loss to the Giants last Saturday.