The deals follow the arrival of defender Kyle Bartley at St Andrew's on Tuesday, on a season-long loan from Swansea City.
Green was out of contract at Mansfield and, despite the best efforts of the League Two club to retain his services, Blues have fought off plenty of competition to land the 26-year-old.
He scored 56 goals for the Stags over the previous two campaigns, including 27 last season as he spearheaded the club's promotion charge back into the Football League.
Centre-back Burn has joined Birmingham on a season-long loan from Fulham, with whom he recently signing a two-year contract extension.
The 6ft 7in 21-year-old spent most of last season on loan at Yeovil, helping them win promotion to the Championship.
Burn told Birmingham's official club website: "I'm really happy to have joined and I'm just looking to get back to pre-season training now.
"The history of this club was something that convinced me to join.
"Lee Clark, having played for Newcastle, is also somebody that I knew a lot about too and there's a lot of good players here. Now it's up to me to go and prove myself."
Burn, who scored Yeovil's winner in their League One play-off final triumph over Brentford, started his career at Darlington in 2009 and joined Fulham at the end of the 2010/11 season.