There is a good answer at the related question What are the pros and cons of installing a roof heating cable? about ice dams and such. I have a dark colored roof and I have dark colored leaf covers (Gutter Helmet, brand). I have a roof snow rake, and I am generally able to keep the ice dams from being a problem. But my gutters and down spouts are white.
When you pull the snow and ice away from the edge of the roof (roof rake), and the sun hits, stuff melts and in my experience on many of those cold sunny days, it drains nicely into the gutters and downspouts where it freezes solid.
If I paint or replace my gutters, would a dark color add sufficient energy to the gutters and downspouts to impact keeping them unfrozen? I looked around online and nothing jumped out at me. It seems obvious that a dark colored item in the sun is going to be warmer than a light colored item, but is there a reasonable chance for dark colored gutters and drains to help with ice issues?