Woody Johnson issued a statement that said a search for a replacement for Tannenbaum would begin immediately.
Johnson thanked Tannenbaum, 43, for his 15 years of service with the Jets. Tannenbaum had been general manager since 2006.
"Although he helped us to two consecutive AFC Championship games, we are not where we want to be and a new general manager will be critical to getting this team back on the right track," Johnson said.
The Jets finished 2012 with a 6-10 record, which was tied for third with the Buffalo Bills in the AFC East. Buffalo beat the Jets 28-9 in Sunday's regular-season finale.
Johnson said Ryan would be back for a fifth season as head coach. The record in 2012 was the first sub-.500 season for Ryan as a head coach. He is 34-30 in four years as the Jets' coach.
"Rex Ryan will remain the head coach of our football team. I believe that he has the passion, the talent and the drive to successfully lead our team," Johnson said.
"Like all Jets fans, I am disappointed with this year's results. However, I am confident that this change will best position our team for greater success going forward."