Unnur started her senior team career with Haukar in the Úrvalsdeild kvenna in 2005 and won the national championship with the club in 2006 and 2007.
[6][7] She returned to KR in 2017, helping the club win the second-tier 1. deild kvenna with a perfect 30-0 season and achieve promotion back to the Úrvalsdeild.
[8] On 6 February 2019, with KR in first place, Unnur Tara suffered an anterior cruciate ligament injury in a victory against Breiðablik and was initially ruled out for the rest of the season.
In KR's second game of the 2019–20 season on 16 October, Unnur, who is a practicing medical doctor outside of basketball, was involved in a controversy with referee Ísak Ernir Kristinsson after she requested permission to enter the court during a stoppage to attend to a teammate who had been suffered a serious leg injury after a hard landing moments before.
[17] In November 2018, she was selected to the national team, following a hiatus of 11 years, for its upcoming games in the EuroBasket Women 2019 qualification.