They also placed safety Brandon McGowan, who started 11 games last season, on season-ending injured reserve with a chest injury.
Burgess, who didn't practice until Aug. 15, was one of 20 players cut by the team as it got down to the regular season limit of 53. He started six of his 16 games with five sacks in his eighth pro season. He has 52 sacks in his career, starting with three seasons in Philadelphia and four in Oakland.
There had been speculation that Burgess, who signed a one-year, $1.5 million contract in May, might retire but he didn't mention that after his first practice this summer.
"I had to take care of my family business before I could report," he said, but admitted he was "a little rusty" in his first practice with the team.
Aiken spent the past two seasons with New England after five with the Buffalo Bills. He had a career-high 20 receptions last season, giving him 47.
McGowan matched his career high with 50 tackles in his only season with the Patriots. He spent his other four with the Chicago Bears.
The Patriots released two other veteran defensive players, linebackers Eric Alexander, primarily a special teams player in his six years with New England, and Pierre Woods, who started eight games in four seasons with the team.
Also released were quarterback Zac Robinson, running backs Thomas Clayton and Chris Taylor, wide receivers Buddy Farnham, Darnell Jenkins and Rod Owens, tight ends Carson Butler and Rob Myers, offensive linemen George Bussey, Ted Larsen, Rich Ohrnberger and Thomas Welch, linebacker Thomas Williams, safety Sergio Brown, cornerback DeAngelo Willingham and defensive back Ross Ventrone.
Some of those players could end up on the Patriots' practice squad.
Among those released were five players taken in the past two drafts — 2009 picks Ohrnberger (fourth round) and Bussey (fifth) and 2010 choices Larsen (sixth), Welch (seventh) and Robinson (seventh).