Fischer in the Arctic
Yes, they do know he has been ranting obscene conspiratorial garbage and generally doing obnoxious things for years.
Iceland every now and then does stuff like that, sort of by accident (like when they accidentally announced that the Baltics should be independent from Russia, as NATO policy). It usually boils down to a small number of people sitting in a hot tub pushing some emotional buttons. In the Fischer case, as before, it is a knee-jerk favouring of an underdog that is being outmatched, and there's this vague "but he made some people take notice of Iceland" warm fuzzies.
This can get us into trouble. Since I think they will resist extradition on whatever grounds the US invents next (I think Fischer was wrong to go play in Serbia in 1992, from what little I know of the circumstances, I also think the US justice department is severly misinterpreting the law in thinking that this was a prosecutable offence; as to whether he owes taxes, well that depends on when his renounciation of US citizenship was effective - when he said or when they said).
Yes, he will turn on Iceland and say something obscenely shocking and insulting about us, I think everyone there knows it and expects it. But insane people should be treated not jailed, the man deserves sympathy and medication, not persecution and prosecution.
So, bottom line, warm fuzzies, sympathy for someone who did us a good turn once, and compassion for the sick. There are worse reasons to piss off superpowers.