Alleyne, 21, becomes Burnley's fourth summer signing after spending the first half of last season on loan in the Championship with Watford, where he made 17 appearances before being recalled in January.
He went on to feature seven times for City's first team in all competitions, including a goal in a 10-1 FA Cup win over Exeter, and has been capped at England Under-21 level.
"I first heard of the interest a couple of months ago now and I'm so glad it got done," Alleyne told the Burnley website.
"It's a huge team, a Premier League team to be honest and I'm so buzzing to be here."
Burnley, relegated from the Premier League last season, begin their Championship campaign at home to fellow-relegated West Ham on Sunday.
Burns, 23, has left Manchester City on a permanent basis, signing a three-year deal at Wigan Athletic. A product of City's academy, his only senior appearance for the club came in an EFL Cup win over Wycombe Wanderers in September 2021, since when he has spent time on loan at Swansea, Stevenage, Hull City and Reading across the past five seasons in search of regular football.
His move to Wigan gives him a fresh start away from City for the first time since progressing through the club's youth ranks.
Both departures reflect City's ongoing efforts to find first-team pathways for academy graduates who have struggled to break into the senior squad on a regular basis, with the club continuing to loan out and sell fringe players as it manages a large development pool under new manager Enzo Maresca.
City have been particularly active in offloading academy defenders this summer as the squad is trimmed ahead of the new Premier League campaign, with both Alleyne and Burns representing the kind of graduate the club hopes can go on to forge senior careers elsewhere even if a path to regular City football never materialised.