I didn't get a chance to on my most recent trip but in January 2016 I climbed the tower for sunrise, definitely recommend for some nice shots.

Time Required: 30 minutes

Cost: 7 GBP to climb tower

Jan 2016 - View from the top of the  Hallgrimskirkja at sunrise

Jan 2016 - View from the top of the Hallgrimskirkja at sunrise

May 2017 - Inside the  Hallgrimskirkja

May 2017 - Inside the Hallgrimskirkja

May 2017 - The  Hallgrimskirkja

May 2017 - The Hallgrimskirkja