The dismissal was automatic based on the school's student-athlete drug policy for a third arrest.
DeKalb County jail records show Rivers was arrested on Wednesday and released on Thursday on four charges, including a felony violation of Georgia's controlled substance act. He also was charged with possession of less than an ounce of marijuana, parking in a disabled parking spot and keeping drugs in a container.
According to jail records, Rivers was released after posting bond of $2,200.
In a statement released by Georgia, coach Kirby Smart said it was "extremely disappointing" Rivers could not take advantage of repeated chances to remain on the team.
"He's been given previous opportunities to remain on our football team but continues to exhibit a lack of good judgment and commitment to the standards we require and expect from our players," Smart said in the statement. "He's put himself in a difficult position but we hope he finds a path that will provide some direction in his future."
Rivers was given a one-game suspension against Kentucky for his first arrest and a four-game suspension in December for his second arrest. The second suspension was to include the first three games of the 2016 season.
Rivers, from Stone Mountain, Georgia, had four tackles in four games as a freshman in 2015, including three against Georgia Tech.