Midfielder Dario Dumic joined Norwich City's academy as a full-time scholar in the summer of 2008 from Danish side FC Hvidore. As a Danish triallist from Himmelev, Dario played for Norwich's Under-13s in Norhalne Cup 2005 (Denmark) in May 2005 and for the Under-14s in the Baisieux Tournament in France in 2006

Dario, who can play as centre-back or in central midfield, was called up in the summer of 2008 to the Denmark Under-17s

Dario signed his first professional contract with the Canaries at the end of June 2009. He made his first team debut as a 57th minute substitute for Jon Otsemobor in the Johnstone's Paint Trophy clash at Swindon on 10th November 2009, which Norwich won 5-3 on penalties.

On 13th January 2010, Dario left the Canaries to return to his native Denmark, signing for Brondby on a free transfer, with payments to be made to Norwich depending on appearances and a potential sell-on clause.

On 19th January 2016, Dario joined Dutch side NEC Nijmegen on loan until the end of the 2015/16 season. He had played 106 times for Brondby, in all competitions, at the time of his transfer. The following month saw the transfer made permanent with effect from June 2016.

He made his senior debut for Bosnia and Herzegovina on 28th March 2017, in a 2–1 friendly win over Albania. His first international goal came against Belgium in October 2017 in a World Cup qualifier.

After NEC's relegation, he was linked with a move to Derby County but signed a four year contract with FC Utrecht on 10th July 2017. He made 18 appearances in his first season with Utrecht.

In June 2018, Dumic signed a one year loan deal with Bundesliga 2 side Dynamo Dresden who have a buy-option in the arrangement.

In June 2019, he was loaned out again, this time to German side SV Darmstadt

In September 2020 he signed for Dutch side Twente and made 29 appearances, scoring twice before joining German second tier side SV Sandhausen  on 4th January 2022. He played 48 games for Sandhausen, scoring five goals.

In August 2023 he moved to Spanish second tier side Eldense who are based in Elda near Valencia.