Wikipedia maintain a list of car-free places, organised by continent and by country (credit to choster's comment). It includes many that aren't islands, but it's easy to find islands on the list. There's also a French-language list restricted to car-free islands on French Wikipedia.
Here's a selection of notable islands taken from that site, I've tried to include only those that are described as completely car free on the entire island, and car-free by policy not simply because no resident has imported a car yet. This is intended for skim-reading for ideas and an overview only, check details on more up-to-date sources:
In Croatia, Koločep (population 120-200) plus three islands that allow service vehicles and tractors.
In Denmark, Ertholmene, Bornholm, population 96
In France, six islands in Bretagne (populations of a few hundred), four in the Archipel du Frioul, near Marseille (total population 86), and Mont Saint-Michel (an island fortress/abbey, population 60)
In Germany, six North Sea islands (populations from around 500 to 2000, check details for each as "residents may be allowed cars on some islands"), the Baltic Sea island of Hiddensee (population 1,200), and Frauenchiemsee in lake Chiemsee (300 residents and a historic abbey)
In Greece, Hydra Island, 50 square kilometers, population 3,000,
In Sweden, the Southern Göteborg Archipelago
In the UK, Easdale, and Herm and Sark in the Channel Islands.
Toronto Islands [Canada]. Several car free islands just off downtown 700 Urban park, neighbourhoods, and ecological community. Accessible by ferry or private boat from downtown Toronto. Bicycles are permitted on ferries.
In the USA: Rock Island, Wisconsin; Tangier Island, Virginia; Russell Island, Isle Royale and Mackinac Island, Michigan; Monhegan Island, Maine; Marsh Island, Louisiana;
In Mexico: Mexcaltitán de Uribe in a lagoon in Nayarit (pop. 818), plus dozens of islands, some densely urbanized, in Laguna del Toro, San Lorenzo, La Santísima, and Xaltocan.
Paquetá Island, Rio de Janeiro, Brazil. Population 7000. Extremely peaceful, car-free island in Guanabara Bay, an hour by ferry from downtown Rio. The place is as quiet as Eden. Residents ride bikes ... never locked and tend to congregate on downtown streets or at the foot of the stairs... In the south portion of the island, Danke de Mattos Park is cycle-free.
(there's also Ilha Grande, Rio de Janeiro, Brazil, population 177,832, but it allows rentable jeeps)
Ilha do Mel, Paraná, Brazil. Small Island offshore of Paranaguá. The Island has two small fishing villages. The entire Island is car free, and even cattle free. All goods and materials are carted around by big wheel barrows. Economy is traditionally fishing but Ecotourism is now a big part of economy. 95% of Island is now an ecological reserve. Villages are connected by 2 meter wide sandy walking paths
The South America section of the list seems to be very incomplete; I know of several car-free zones not listed - if you're interested in South America, don't be deterred by the few entries!
- Lamu, Kenya, population of several thousand
Lamu town is an old, Swahili settlement where only foot, cycle and donkey traffic is allowed.
Like South America, the Africa section appears to be very incomplete.
Asia & Oceania
Four islands, Büyükada, Heybeliada, Burgazada, Kınalıada, off Istanbul in Turkey, where only emergency service vehicles are allowed.
Perhentian Islands in Malaysia (x2 islands), and Gili Islands (x3), off Lombok, Indonesia, both popular tourist attractions with populations of several thousand.
Phú Quốc, off the town of Hà Tiên in Vietnam. Population 90,000. largest island in the Gulf of Thailand. Officially car-free, and the regulation is largely respected.
Cheung Chau, Hong Kong [China]. Population 30,000. No full size automobiles allowed. A few miniature emergency vehicles exist as do small utilitarian vehicles. Residents walk, via an extensive and well-maintained network of trails. Ferries take residents to the rest of Hong Kong.
- In Australia, Rottnest Island (Western Australia), French Island (Victoria) and Maria Island (Tasmania) have no cars allowed and low populations (although Rottnest has buses)