Feldman was due $9.25 million. Instead, the team will pay him a $600,000 buyout. Tateyama had a $1 million option.
Feliz, 24, had season-ending Tommy John surgery on his right elbow Aug. 1. He was a member of the Rangers' Opening Day rotation, going 3-1 with a 3.16 ERA.
Texas still can sign Feldman and Tateyama, but with their options declined, both become free agents.
Feldman, 29, signed his original deal prior to the 2010 season, after he went 17-8 in 2009. He was the club's Opening Day starter in 2010, but he was never able to replicate the 2009 season.
A knee injury that required surgery limited his 2010 and 2011 seasons. He did have some success in a long relief role, but was not able to keep a rotation spot.
Feldman was 6-11 with a 5.09 ERA in 29 games -- 21 of them starts -- this season. Tateyama allowed 17 earned runs in 17 innings in three stints with the Rangers this season.