ShotHotspot has mapped 20 abandoned places in Iceland, ranked by how often they're photographed, viewed and voted for. The most popular are Plane Wreck Dc3 Dakota, Lighthouse in Akranes and Seljalandsfoss. Some are private property or unsafe, so get permission before you visit.
We don't know whether you're allowed to visit any of these places. Many are private property, closed to the public or unsafe: get permission from the owner or the relevant authority, and take safety precautions.