The 33-year-old had a stellar 2012 season for the Twins, posting a 2.18 ERA in 64 appearances with an opponent batting average of .186. But he regressed the last two years, with his walks going up and his strikeouts going down.
Burton was 2-9 with a 3.82 ERA in 71 games in 2013. He went 3-5 with a 4.36 ERA in 68 games this season while losing his spot as the primary setup man.
Minnesota announced its decision Wednesday.