Dominica has two airports, so the place isn't devoid of civil aviation.
Photo © Isa Ali
Photo © Patrick McManus
Liechtenstein has no airports, but there is a heliport, and looking at the photo database, there are some aircraft registered under the Swiss HB register. I guess if Switzerland administers civil aviation there, then there is no reason for the country to be part of ICAO.
Photo © Rolf Wallner
Photo © Thomas Schmid
Tuvalu has an airport, but I don't believe it has any aircraft , and is only visited by an Air Fiji service.
Photo © Ward Callens
Montenegro has only recently become independent of Serbia, and was admitted to the UN just last week, so presumably it will soon be a member of ICAO.
Meanwhile, the Cook Islands is self governing in free association with New Zealand. As such, New Zealand is responsible for its foreign relations, and it is not a member of the UN in its own right. It is however now a contracting state to ICAO, and has its own registration prefix (E5). It appears to be the only non-UN state that is a contracting state to ICAO.
Photo © Richard Greaney
Does anyone have any more information about why Dominica, Liechtenstein and Tuvalu are not contracting states, and why the Cook Islands is?