The move clears space on the roster for recently-acquired Trevor Davis, who comes over in a trade from the Packers.
Burris, 26, is a former fourth-round pick of the Jets back in 2016 out of NC State. He was in the third year of his four-year, $2.89 million contract when the Jets waived in October of 2018.
The Jets re-signed Burris to their practice squad after he cleared waivers but was signed by the Browns off their taxi squad after a week.
Cleveland elected to decline to tender a qualifying offer to Burris this offseason, but later re-signed him to a one-year, $645,000 deal back in April. He was among their final roster cuts this offseason.
In 2018, Burris appeared in eight games for the Jets and Browns, recording two total tackles and no sacks.