New Orleans experienced the same wave of nineteenth-century immigrants that swelled the East Coast--from Ireland, Germany and Italy.This can't be the explanation, for the simple reason that plenty of other places (Chicago, for instance) experienced the same immigration, but not the sound change. Oddly enough, a couple of months ago the New Yorker quoted no less an authority than William Labov observing, rightly, that although the New York dialect is thriving, it is losing this very feature (as well as a few others I'll write about later). Interviews in the wake of the hurricane demonstrated that in New Orleans this feature is robust. Setting it further apart from its northeastern counterpart is the fact that in New Orleans this feature has no racial dimension (think of Louis Armstrong singing "What A Wonderful Woild"), whereas the now-dying New York area "oi" never was part of the local African American vernacular, even though it has incorporated other distinct New York features.
So where does this strange feature come from? My guess, and I don't think I'm too far out on a limb, is that it is what is known as retroflex, or rhotacized, vowel coloring. That is, in both New Orleans and the New York area "r"s are dropped after vowels, like in many other kinds of English, and in some -- not all -- of the dialects where this happens, it alters the preceding vowel. I think it's jut an odd parallel development in these two non-rhotic ("r" dropping) urban dialects that can be explained without recourse to history, aside from the forces that led to derhotacization throught the Eastern Seaboard and Gulf Coast.
I have to admit though that my gestalt impression of New Orleans English is that it is strongly suggestive of an East Coast city, with a whiff of southernness. Sort of like Philadelphia.