The problem here, and I've discussed this elsewhere, is that we're not really sure who actually saw themselves as "Sumerian" or what that meant to them. We can say with some confidence that the Sumerian language was the lingua franca of southern Mesopotamia(albeit with considerable overlap between it and Akkadian) until it became increasingly marginal sometime around the mid/late 3rd Millennium BCE and dying out completely as a spoken language sometime around the end of the 3rd or beginning of the 2nd millennium(the exact timeframe is heavily disputed and difficult to pin down, 2000 BCE is a useful ballpark with most scholars leaning somewhat earlier but good arguments for leaning later).

What we still don't understand is how this linguistic shift interacted with ethnic self-definition; we have very few references to people actually being described as Sumerian(or for that matter as Akkadian) and it's hard to suss out what people meant by that. Having said that, I'm not aware of any good evidence for wholesale population replacement in southern Mesopotamia during the literate period. In other words, trying to pin down a "Sumerian ethnicity" is borderline impossible(and frankly a product of a lot of 19th century history and anthropology's occasional or regular obsession with racial categorization) and we can hardly speculate on if or how it "disappeared" when we don't really know what it was.

If you want to ask how Sumerian died out as a spoken language* we are on firmer but still very badly documented ground. We do know that there was a substantial period where Sumerian and Akkadian coexisted and bilingualism was widespread because there are a large number of Sumerian loanwords into Akkadian as well as some grammatical peculiarities of Akkadian that are usually explained by contact with Sumerian. It also probably followed similar patterns to other linguistic changes where moderate population shifts and changing social patterns(political changes making for a new prestige language, say) make it become increasingly necessary to be bilingual leading to the less-used language dying out as fewer and fewer people use it in fewer and fewer contexts. But Sumerian culture still remained important and in many ways a foundational reference point for much of Mesopotamian history.

*Sumerian stayed in use as a literary and later a cultic and ritual language long after it died out in speech communities; in this respect it was somewhat like Latin in medieval Europe.