The 34-year-old received an 80-game suspension on March 6 after testing positive for a performance-enhancing substance during an offseason test.
He lost his appeal and he is ineligible to play in any postseason games this year.
In January, Wright and the Red Sox agreed to a one-year contract for $1,375,000, a raise of $275,000, in a deal that avoided arbitration.
Wright was 3-1 last season with a 2.68 ERA in 53.2 innings pitched for the World Series champions.
The team also announced that right-hander Nathan Eovaldi was moved to the 60-day injured list. The 29-year-old was originally put on the 10-day IL with a loose body in his right elbow on April 20, retroactive to April 18.