The Barnsley-born Second Row joined the Falcons from Championship side Yorkshire Carnegie in summer 2013 on a two year deal and last year signed a further contract extension at the club.
However, having made 37 appearances for the North East side Barrow, 22, has chosen to exercise a clause which allows him to terminate his contract earlier should the Falcons finish below tenth in the league.
Falcons Director of Rugby Dean Richards said: “We’re bitterly disappointed at Dom’s decision to leave but unfortunately despite our best efforts, discussions with his agent proved very difficult.
“As a result Dom has left us to join Leicester Tigers, where he will complete rehab following his ACL repair.
“Our Academy has produced a number of talented locks in recent years including Will Witty and Sean Robinson, both of whom made their Premiership debuts this season and will now have the opportunity to make an impact in the starting line up during Rugby World Cup at the start of next season.”