Tuttle replaced Colin Lee in December, but leaves the club after winning just four of his 20 games in charge.
Tony Burns and Alan McLeary will take over the first-team for the final two games of the season.
He said: "Naturally I am very disappointed we were not able to avoid relegation, but I would like to thank the staff and players for giving it their best shot.
"In particular, I would like to thank the fans for the way they have supported the team during a very frustrating season and I wish the club every success for the future."