AFC Bournemouth have signed Christian Atsu from Chelsea on a season-long loan deal.
Midfielder Miguel Pallardo has signed a new one-year deal with newly-promoted Hearts while Danny Wilson leaves.
The 30-year-old has moved to the City Ground after rejecting the chance to extend his three-year stay with Birmingham City.
Burke made 155 appearances for the Blues and scored 28 goals after joining them from Cardiff City on a free transfer in June 2011.
He was with the Bluebirds for two-and-a-half seasons after heading south from Rangers, for whom he made his debut as an 18-year-old in March 2002.
Burke, who has made seven appearances for Scotland, spent seven seasons at Ibrox and featured in UEFA Champions League games for the club.
After putting pen to paper for Forest, Burke told the club's official website: "I'm delighted to be here.
"Forest have always been a club I've enjoyed playing against. The City Ground has a great atmosphere with great fans and I'm looking forward to being a part of that now.