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
May 2017 - Inside the Hallgrimskirkja
May 2017 - The Hallgrimskirkja