You can ride anywhere there’s a track. But not off the tracks. The landscape is incredibly sensitive, and tyre tracks could still be there in tens of years time.
In Highlands are definitely the place to go. While it’s pretty wild, there are lots of huts, there’s lots of fresh water and the going is typically okay (though the corrugations can get a little tiresome).
This is a good resource for what tracks are open when:
Mountain Roads
This is an exceptionally detailed map of the country, you can seet tracks which aren’t on most maps: Iceland Mapping
Ferdakort make pretty much all of Icelands paper maps. They’re excellent.
This map is also great, but the info on fords is a load of lies! There are many more and some are… urm… interesting: Map of Lies