Tate has spent six of his eight CFL seasons with Calgary and last year scored a league-high 10 rushing TDs as the Stampeders' backup quarterback.
The 30-year-old rushed for a pair of touchdowns in the club's 20-16 Grey Cup victory over the Hamilton Tiger-Cats in November.
Tate and starter Bo Levi Mitchell both signed contract extensions in January, 2014. Tate started four games and won three last season when Mitchell was sidelined with a leg injury.
Tate completed 66-of-111 passes for 823 yards and five touchdowns and added 129 yards rushing on 42 carries for the 10 TDs.
He was tabbed to be Calgary's starting quarterback in 2012 when the Stampeders dealt veteran Henry Burris, but shoulder and arm injuries have limited Tate's playing time the past two seasons.
Calgary released Lemon in order to allow him to sign an NFL contract.
Lemon and the agency that represents him announced on Twitter that he's signed with the Pittsburgh Steelers. He was scheduled to become a CFL free agent Feb. 10.
Lemon, 26, had a club-high 13 sacks last season and tied a league record with eight forced fumbles.