Darlington-born Burn has been with Sunderland since he was 13, and was in the second year of a scholarship.
He spent time on trial at Boro earlier this season, and now they have agreed to take over his terms - which run until 2014.
Burn spent time on-loan at Northern League Darlington earlier this season.
Meanwhile, Boro have confirmed eight of their own young players will be released in the summer.
They are goalkeepers Shane Bland and James Montgomery, defenders Paul Weldon and Kyle Oliver, midfielders Curtis Edwards and Jarrett Rivers, midfielder-striker Dylan McGlade and striker Adam Garrity.