Jensen, the Clarets' longest-serving player after moving to Turf Moor from West Brom in June 2003, will become a free agent at the end of his testimonial year.
The 37-year-old, affectionately known as "the Beast", made over 300 appearances for Burnley and starred during their Premier League season of 2008/09.
He made an emotional final appearance for the club late in the last game of the season against Ipswich at the weekend, replacing the ever-present Lee Grant and receiving a rapturous ovation from the Turf Moor crowd.
After being informed of the club's decision not to offer him a new deal, Jensen tweeted: "Emotional day. Not been offered a new contract so time for a new challenge.
"I want to thank all the fans for supporting me over the years. Much appreciated. Need you one last time at the testimonial game."
A statement on Burnley's official website, burnleyfootballclub.com, said: "Burnley Football Club would like to thank Brian for his invaluable contribution to the club in the past 10 years and wish him and his family all the very best for the future."