There is a hypothesis that Sumerian was somehow connected to the Dravidians of southern India, i.e. the Indus valley civilization. Is there any concrete evidence on this or just a loose theory based on proximity?
The origins of Sumerian are one of the hardest problems in historical linguistics because its right at the beginning of the written record, because the writing system is not very phonetic, and because it can't be showed to be related to any known language. As Sumerologist Abraham Hendrik Jagersma put it in 2010:
More crucially, Sumerian is a language isolate and it is highly unlikely that we will ever find a language related to it. ... the linguistic landscape of which Sumerian was a part is largely unknown and has long since disappeared, mostly without a trace. Linguistically, modern Iraq is three language shifts removed from Sumerian: from Sumerian to Akkadian to Aramaic to Arabic. Similar language shifts have happened in neighbouring Iran, Turkey, and the rest of the Middle East. Any traces of related languages have thus been obliterated thousands of years ago.
Its plausible that Sumerian had relatives which died out without being written down, but for a language to have no relatives was probably not unusual in 2500 BCE: the expansion of the Indo-European, Afto-Asiatic and Turkic language families can all be linked to special circumstances (the spread of agriculture, horse-borne warriors, and imperialistic states). As linguist Don Ringe argued in 2009, before the spread of Indo-European there might have been on the order of 30 languages in Mediterranean Europe, in 15 or 20 families compared to the eight families and 22 languages (plus Basque and Punic) which are attested in texts before the year 1.[1] In places like the Americas, Australia / New Zealand, or highland Southeast Asia that didn't get conquered and assimilated by horse-borne warriors trailing bureaucrats until a few hundred years ago, we see lots of isolates and families of two or three languages.
So the people with theories on the origins of Sumerian tend to be a mix of senior professors and non-linguists elbowing their way into the room, and while some of them are very impressive they are not very good at convincing other people. Two papers by researchers at universities in Europe which discuss the problem and propose theories are:
But like most things with Sumerian, if you ask three experts you will get four different answers.
I have heard theories that Elamite (spoken in modern Khuzestan and Fars in south-west Iran) is related to Dravidian, but I'm not familiar with the languages and the research to give my thoughts on that theory.
[1] Ringe, Don (2009) "The Linguistic Diversity of Aboriginal Europe." Language Log, 6 January 2009 https://languagelog.ldc.upenn.edu/nll/?p=980 His list does not include Rhaetian or Celtic, but its a freely accessible blog post not a carefully written article.