I've recently discovered that in the American magazine The New Yorker, as a matter of in-house style, a diaeresis is used whenever a pair of vowels forms two syllables where it might otherwise be expected to be diphthongal. Thus, flicking through its pages you might come across the story of a “preëminent” politician requesting “coöperation” from his colleagues. My own writing convention has been to use a hyphen to separate identical vowels but otherwise to leave well alone—hence, “cooperation” becomes “co-operation”, while “coincidence” (as luck would have it) stays as it is—but as someone with both a certain fascination with language and an affection for the esoteric, The New Yorker's convention holds quite an appeal.
Diacritic marks—also known, not strictly accurately, as accents—like the diaeresis, are of course used in the written forms of many languages for phonetic clarity and semantic disambiguation. Sometimes these two aspects go together; sometimes they don't. In Spanish, for example, the acute accent in está (it is) indicates a difference in stress from esta (this), whereas cual (which) and ¿cuál? (which?) are pronounced identically. In the latter case the diacritic is simply a visual indicator to differentiate the relative and interrogative meanings. Of course when spoken, intonation will tell them apart.
It is generally assumed by most English speakers that their language makes no use of diacritics at all. However, English words have a great mixture of origins—roughly equal parts Germanic (Anglo-Saxon), French and Latin, plus bits and pieces—and diacritics from the source language often get dropped (as in “cafe” and “role”) without the spelling of loan words being completely anglicised. The word “apostrophe”, which comes from Greek via French, is a direct transliteration from the Greek ἀπόστροφος, whereas the Spanish equivalent apóstrofe uses a native spelling. Without the clarifying benefit of diacritics, the result is a hellishly unpredictable many-to-many relationship between spelling and pronunciation, to which anyone who has learnt English as a second language can attest. So, what of the diaeresis? As I understand it, the convention came over the English Channel from France along with so many words; but in any case, it was quite common to write this way in English until the early 20th century. The New Yorker certainly did not invent it.
In fact, it is worth noting that the diaeresis is not the only diacritic native to English. The grave accent can be used to indicate that the “-ed” suffix should be pronounced as a separate syllable where otherwise it would not be, as in “agèd”; and an acute accent is occasionally added to indicate that an otherwise silent “e” should be pronounced, as in “saké”. Both of these examples would be ambiguous without the diacritics, since the meanings of “aged” and “sake” are different.
It may be tempting to dismiss the diaeresis convention as nothing more than an historical curio that a sentimental journal has arbitrarily and artificially chosen to preserve. This is not wholly true. Once I had discovered the convention, I began to realise that this was something I had come across without really noticing it. Perhaps I had tacitly assumed that the name “Zoë” and the word “naïve” were simply foreign in origin, and left it at that. Indeed they are foreign—the former is from Greek while the latter is French—but the use of the diaeresis in English is entirely consistent with the New Yorker convention. Certainly both words are often spelt without the diacritic, but the fact that my spell checker hasn't drawn a red line under these versions shows that the phenomenon cannot be considered obsolete.
Occasional surviving examples aside, the use of diacritics in modern English would generally be considered archaïstic (as I can't resist to write)—that is, a conscious imitation of old fashioned orthography. But I see no reason that accusations of pretension should be a barrier to usage of a truly helpful convention. The diaeresis is positively useful both for disambiguation (e.g. “coöp” vs. “coop”) and for aesthetics in words with multiple prefixes (the spelling “uncoöperative” is far more pleasing than either “unco-operative” or “un-co-operative”). Any rule that makes English clearer and more consistent is welcome in my book. If it's archaïsm, so be it. I like it; and apparently so does the New Yorker.
Resources
The New Yorker and its Wikipedia page.
The diaeresis, from Everything2.com.
Source languages of English words, from Oxford Dictionaries.
English language, from Wikipedia.
Accents and other diacritical marks in English, by Scott Horne, a Canadian translator.