South Georgia has no nationals and no permanent residents. I also noticed a couple of pacific islands included that do not have electricity. British Virgin Islands is an important relay station, there are always techies there.
Rarest passport stamps

join team north korea https://www.chess.com/club/north-korea-national-chess-team/join

I have Vatican City, Zimbabwe and zambia in my passports, and I didn't know it until I checked my passports and went down. I was dumbfounded when I saw them.

Ig I need to play with a Fijian, French Guiana-n, Ghana, Greenland, Cote d'Ivoire, Isle of Man, Laos, Liechtenstein, and Nauru (although I did play someone from this country but achievement doesn't show up)

North Korea should theoretically be the rarest one, but I think people realized that and started putting it as their country as a joke, so it went up. In fact, I think I've had 4 or 5 "North Korean" opponents in my time.
If we were to suddenly bring attention to the other ones that are almost impossible as well (see the above replies, for example), I'm sure they wouldn't be so "rare" either after a while.

In general, all the Island nations are rare, many of the Caribbean Islands were my last to get and I am still missing 2, Monserrat and Anguilla. Additioanally, I am missint the island nations of Northern Mariana Islands and Niue. The only other 2 I am missing are Olympic and Paralympic. When did they add these, this is the first time seeing these.
I think Zimbabwe is the rarest in my opinion! I got one of those. I think you just get the passport after a player match when I think the player doesn't live from that place. Cuz people can change their flags
I have Zimbabwe stamp