Sumer was a city-state in Mesopotamia. It appears to end about 1900 BCE as a political power, as other peoples ("Amorites") push into the region. Sargon of Akkad achieves lasting fame as perhaps the first "Emperor" in history; he conquers and rules what had been Sumerian polities. The Sumerian language and writing system continued to be used, but ultimately Mesopotamian populations become dominated by Akkadian; the last cuneiform tablets using the Sumerian writing system (but to write Akkadian) are roughly the first century A.D. See

Cooper, Jerrold S. “Sumerian and Akkadian in Sumer and Akkad.” Orientalia, vol. 42, 1973, pp. 239–246

. . . for a more extensive discussion of the linguistic evidence for the interchange between Akkadians and Sumer; this was a complex society using both languages, sometimes bilingually, sometimes for different purposes, for a thousand years or more. As a parallel, you might think of the way that Latin and other languages functioned side by side in Europe, long after the Roman Empire itself had disappeared. Sumerian is to subsequent Mesopotamian civilization a bit like that-- there's a language, a writing system, these persist long after the particular city state they belonged to has lost power.

Population genetics suggest that some people of Iraq have ancestry in the region going back many thousands of years -- notably the "marsh Arabs"; that is not definitive proof that they are descended from Sumerians-- but they have been in the region for thousands of years, and their genes show a signal not descended from more recent arrivals to the region (eg Semitic peoples, South Asians). This is a domain where the molecular geneticist's toolkit rather than the historian's provides the answers, see:

In search of the genetic footprints of Sumerians: a survey of Y-chromosome and mtDNA variation in the Marsh Arabs of Iraq for an extensive discussion.

So -- if there's a descendent of ancient Sumer alive today, the best place to look for him/her would be very close to the site of ancient Sumer itself, in the marshes around Basra in Iraq, just north of Kuwait.