Birkir started playing football with local clubs in his hometown of Akureyri. His family moved to Norway in 1999 and he joined the youth teams of Figgjo. In the summer of 2005 Birkir joined Viking. In the 2006 season he scored his first goal for Viking against Hamarkameratene. Following that he played consistently as a starter for Viking and was a key aspect in helping the club avoid relegation that year and helping them obtain a third-place finish in 2007 under trainer Uwe Rösler.

In 2008 Birkir had a successful loan spell at newly promoted Tippeligaen side Bodø/Glimt, playing a big part in them finishing fourth in the league. His performances brought the attention of bigger clubs such as Serie A club Reggina during January 2008, but he stayed at Viking until 2012. During this time he made 100 appearances for Viking in the Tippeligaen, scoring 16 league goals.

Birkir made his debut for the senior Iceland national side in 2010. He has also represented them at Iceland U-21, Under 19 and Under 17 levels.

On 12 January 2012, it was announced that Birkir would sign for Belgian Pro League side Standard Liège on a five-year deal. He played 16 league matches in the Belgium Pro League for Standard.

After the end of the 2011–12 season, in July 2012, Birkir joined the newly promoted Serie A side Pescara on a season-long loan deal, for €300,000, with an option to sign for an additional €900,000. He scored his first goal for Pescara in a 5–1 away defeat at Napoli. After the 2012–13 season he stated that he was not interested in going down to Serie B with Pescara and, with two years remaining of his Standard Liège contract, he was unsure where he would be playing in the 2013–14 season.

He scored his first goal for Iceland against France on 27 May 2012 in a 3–2 defeat. He also followed this up with goals for Iceland in 2014 FIFA World Cup qualification with goals against Albania on 12 October 2012, Slovenia on 7 June 2013 and then against Albania again on 10 September 2013 in a 2–1 victory. Birkir’s goals and performances played a crucial role in the qualifiers helping guide Iceland to the World Cup playoffs, however in November 2013 they lost 2–0 on aggregate over two legs to Croatia.

Pescara bought Birkir from Standard Liège on 9 June 2013 on a permanent transfer for a previously agreed fee, with the aim of selling him for a higher amount that summer with interest being reported from clubs in the top leagues in England including, Italy and Germany.

On 2 September 2013, it was confirmed that Serie A side Sampdoria had signed Birkir on a co-ownership deal. In total he made 14 Serie A appearances during the 2013–14 season for Sampdoria.

On 20 June 2014, it was announced that Birkir would sign again for Pescara on 1 July 2014, after the club won the co-ownership bidding for the player from Sampdoria. Upon signing, he was appointed the captain for the 2014–15 season.

He scored 12 goals in 38 Serie B matches to help guide Pescara to a seventh-place finish and qualifying for the Serie B playoffs. Pescara beat Perugia 2–1 to qualify for the playoff semi-final. They beat Vicenza Calcio 3–2 over two legs in the semi-finals, with Birkir scoring the crucial match winning aggregate goal for Pescara on 2 June 2015 to help them qualify for the Serie B playoff final against Bologna. However, with a 1–1 aggregate score after two legs Bologna, being the highest-placed team, were promoted.

On 27 June 2015, Pescara announced they had accepted an offer of €1 million for Birkir from Serie A side Torino. It was later revealed in July, that the move had seemingly stalled as Birkir had been unable to agree personal terms with the club. A move to English side Leeds United also fell through.

On 7 July 2015, it was announced that Birkir had signed a three-year contract with Swiss Super League club FC Basel for €1 million. He made his first team league debut on 25 July 2015 in the 3–2 away win against Grasshoppers. In the Champions League third qualifying round on 5 August he scored his first goal for his new club during the 1–0 home win against Lech Poznań.

On 27 June 2016, Birkir played against England in the UEFA Euro 2016 round of 16 at the Stade de Nice, as Iceland upset England with a 2–1 victory to advance to the quarter-finals.

On 25 January 2017, Birkir signed a three-and-a-half-year contract with English Championship side Aston Villa. He scored his first goal for Aston Villa in an EFL Cup tie against Wigan Athletic on 22 August 2017.

In May 2018 he was named in Iceland’s 23-man squad for the 2018 World Cup in Russia.

On 18 January 2020, he returned to Italy, after signing a 1.5-year contract with Serie A side Brescia, after a short spell in Qatar playing for Al-Arabi.