Well, for major landmasses, there aren't many. You're looking for big islands, and Madagascar and Iceland were over a thousand years ago. New Zealand was settled around AD 1200 - 1400 roughly, so that counts, and that's it for "big" (>100,000km) islands. Now I'm a little unsure on how and when the various Canadian islands were settled, but it appears many of them were known to the Inuit for a while, so I'm not sure how to date the first occupation. ANybody know anything about this?

So, by 1400AD every continent but Antarctica, the wold's large islands, and most of Polynesia was already settled. Thus we're down to small islands. This is by no means a complete list, just the ones that came to mind:

Easter Island may have been settled post 1000AD, but my understanding of Polynesian settlement dating is that things are kind of in flux, so maybe a little earlier.

Elsewhere: Svalbard (I'm not exactly sure. A quick search on wikipedia suggests 1630 for first year round inhabitants, but I swear I read somewhere it was earlier. Definately post 1000 though); Madeira and the Azores (15th-ish, I've read a few theories about earlier inhabitants, but I don't find them credible, feel free to correct me); Cape Verde (1462); São Tomé and Príncipe (15th); St. Helena (16th Century); Bermuda (17th Century); Mauritius (17th, abandoned, again in the 18th permanently); Réunion (17th) Falklands (definitely post 1700, depends on how you cont settlement); Ascension Island (18th); Chagos Archipelago and Diego Garcia (18th? Maybe, depends on the island, and how you define settled); Juan Fernández Islands (18th, now abandoned, for seals, pirates, a penal colony) South Georgia and the South Georgia Islands (19th century, various islands were settled for whaling, now abandoned, still have a few scientific missions); Lord Howe Island (19th); Christmas Island (19th); Cocos Island (19th, incidentally the first permanent settlement was a former British Governor of Malaysia and 40 malay wives in a strange harem set up that quickly fell apart); Baker Island, Clipperton Island, (19th for Guano, now abandoned); Johnston Atoll (19th, abandoned in the 2000s after the US Military left) The Galapagos (19th, a settler colony) Tristran de Cunha (19th Century); Kerguelen Islands (attempts made in the 19th century, permanently inhabited by a French scientific team post 1950);

This is by no means an exhaustive list (I missed a bunch of Arctic Islands) and sort of depends on what you count as "occupied." Do Shipwrecks count? If they stay for a year, then leave? What about military? Or people who come every summer and then leave for the winter? and so on.

This isn't an answer to your question, rather it's a slight addition or correction:

I learned only in the last year or so that the first human inhabitants of Iceland arrived likely only in the century or so before the first Norsemen. I was kind of surprised at the lateness of this.

Actually the general consensus among historians today is that Norsemen and their Celtic slaves were probably the first to arrive. The legend of Irish monks settling in Iceland is probably (but not certainly) not true. It appears to be mostly based on anecdotal evidence (place names) and the writings of Ari fróði Þorgilsson which is hardly an accurate source given that Ari lived almost 300 years after the settlement of Iceland.

Of course, there are some who disagree and believe that Irish monks were the first settlers, but that's becoming a rare view among historians.

(Source: BA in history from the University of Iceland - I'll go dig out some proper sources if people want.)