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You can enter Switzerland on your national ID, however you are not allowed to use your driving license. As per the website of the Swiss State State Secretariat for Migration Italian citizens can use a valid national identity card to enter Switzerland, however there's no mention of a driving license. As such you would have no issues to exit the airport to visit Geneva.
As for Iceland, according to the Icelandic Directorate of Immigration website, you can enter with a Carta d'IdentitΓ with again no mention of a driving license.
As others mentioned in the comments, both Switzerland and Iceland are part of the Schengen area, even though neither is part of the EU. As such you might not even encounter an immigration check during the trip.