Defender Wilson, 30, made 51 appearances after signing from Celtic in 2013 for a second spell at the club.
Former Hamburg midfielder Tesche, 28, signed as a free agent in August 2014 and featured in 50 games for Forest.
Winger Burke, 32, scored six goals in 54 games after joining in July 2014 following his release by Birmingham.
Managerless Nottingham Forest finished 16th in the Championship this season, 15 points above the relegation zone.
Nottingham Forest have announced Kelvin Wilson, Robert Tesche and Chris Burke will be released at the end of their contracts in June.