The Tempest is one of those rare Shakespeare plays whose plot is not based on a known source. However, in Act 1, scene 2 Ariel says to Prospero:
Thou call'dst me up at midnight to fetch dew
From the still-vex'd Bermoothes
This implies that the island is not to be identified with any of The Bermudas, which are not on the way from Tunis to Naples anyway.
However, many scholars think that the storm and (apparent) shipwreck in the first scene of the play were inspired by a real event. In 1609, a fleet of nine ships was sailing to North America when it was caught in a ‘terrible storme’ near Bermuda. One ship sank. Another, the Sea Venture or Sea Adventure, was wrecked on the coast of the Bermudas. One of the survivors of the storm, William Strachey, wrote an account of it, which wasn't printed until 1625 but which already circulated in manuscript in 1610.
Strachey writes that the Bermudas were commonly called "The Devil's Islands" and that they
are feared and avoyded of all sea travellers alive, above any other place in the world.
One topos in accounts by travellers to the New World was cannibalism (see Orgel, 33–34). Shakespeare probably found an account of cannibalism on Montaigne's essay "Of the Caniballes" in John Florio's translation of Montaigne. Texts like this may have given Shakespeare the idea of naming one of his characters Caliban. However, Montaigne's essay is not based on cannibals living on an island.
Based on what I have read so far about The Tempest, looking for a real island will probably not teach us much about the play (unless scholars dig up a source that was hitherto unknown). It is more interesting to look at how Shakespeare used his sources, including the ideas and worldviews represented in them. In the introduction to his edition of The Tempest Stephen Orgel points out (page 35; emphasis mine) that
Caliban has almost nothing in common with the prelapsarian savages described in Montaigne's essay "Of the Cannibals", (...). He owes more to the concepts of the natural depravity of the New-World populations, such as are found in explorers' accounts from Purchas to Captain John Smith. In Montaigne, however, it is the Europeans who are predatory and savage; Shakespeare, as he so often does, dramatizes both sides of the debate, and in the process renders a solution to it impossible.
What Shakespeare does with his sources of inspiration is usually more interesting than the sources themselves, many of which would have been forgotten without their connection with Shakespeare.
 Shakespeare sometimes seems to take liberties with geography. In All's Well That Ends Well, a character travelling from Roussillon to Santiago de Compostela ends up in Italy. And in The Winter's Tale character is eaten by a bear on the seacoast of Bohemia.
- Shakespeare: The Tempest. Edited by Peter Hulme and William H. Sherman. Norton Critical Editions. Norton, 2004. (Pages 110–115 reproduce excerpts from Strachey's letter.)
- Shakespeare: The Tempest. Edited by Stephen Orgel. The Oxford Shakespeare. Oxford University Press, 1987. (Appendix B contains Strachey's letter; appendix D contains excerpts from Florio's translation of Montaigne.)
- Strachey's 'A true reportory of the wreck' in Bermuda, The British Library.