The diaeresis potentially contrasts words; it indicates a syllable boundary, and prevents hybrids like ‘zoology’, where the vowel of the first syllable has been infected by the 'oo'. (For actually contrasting words, it’s vary rare - ‘coöp’ v. ‘coop’ is the best I could come up with quickly, and the former is normally spelt ‘co-op’.)
This can be done in principle, but spellings like "reëlect" and "coöperate" can descriptively be regarded as a New Yorker–only quirk, absent from ordinary writing, even if confine ourselves on the "educated prose" register.
I regularly see "naïve", but "naive" is just fine, like the other diaeresis-less variants. Very rarely one encounters "über-", but (as most people here will know), there it has a different function (and in German we btw don't call German ¨ a diaeresis or trema but just don't give it a name, {ä,ö,ü} being regarded as separate letters).
Other English words where inclusion of a foreign accent is somewhat more frequent (but not "descriptively obligatory", as far as I can tell and this can be judged) are very rare. Candidates are "fiancé(e)", "façade", "soupçon"; "café" already a bit less so I'd say. And, they're clearly all loanwords. Many of these can be regarded as older spellings. Then there are some (not-that-frequent) foreign words, but they can often be said to not be part of non-technical, vernacular vocabulary. If a culinary item has many accents, it's partly because of where it's occurring.
Stephan
