Kuivasaari: it is situated near to 8 km southward from Helsinki harbour. It is a military zone but there are excursions yearly usually in the end of August (cost 25 euros for 2-3 hours).
Suomenlinna: see here.
Isosaari: it is in 1.5 km eastward of Kuivasaari island. Island is restricted military zone too.
Santahamina: this is the big island not far form Helsinki. It could be reached by car (bus) from Helsinki through the dike, but there is a check-point on the dike, because the island is military zone.
Russarö: it is near to 5 km southwards Hanko (where Hanko is see here). It is in restricted zone, of course.
Örö: it is near to 30 km westwards from Hanko. There are the excursions organized from Kasnas city for 50 euros cost (see in the Links).
Bolax: as far as 17 km north-west from Hanko harbour. You will need to hire a motorboat in Hanko for this adventure.
Ristiniemi (Krestovy) Cape: it is in the border zone now and it is not an easy to get there for foreigners especially. There is 1-1.5 hour drive from Vyborg - Torfjanovka (Valimää) road and 1-1.5 hour walk.
Tuppura (Vikhrevoy) island: it is in 2-3 hours motor boat sail from Sovetsky settlement (Jensuu) not far from Primorsk (Koivisto) city. It was not in the border zone in 2005, but I'm not shure about present time.
Puumala: it can be easy reached by the bus from Zelenogorsk (Terjoki), but the place has been fenced (summer 2008) and it could be a problem. One of the MU-2 battery can be find here too.