No.
I mean, it depends on how you define 'oldest language in the world'. Does that mean 'the first language ever'? Oldest language we have evidence for? Does that mean 'the oldest one that is still in use'? And if so, because languages evolve and develop into new ones, how do you define the boundaries between old languages and their immediate descendants?
However you answer those questions, though, Tamil still isn't the oldest language in the world.
Before I proceed, I'm gonna first point you to /r/linguistics. Their FAQ has a section on historical linguistics, including things like why we can't easily say what the oldest language is and what's the deal with Tamil. I'll also give a shout to our FAQ section on language, just cuz.
I'm not gonna get too into the weeds on why Tamil isn't the oldest language. As I demonstrate in this answer, there isn't enough evidence out there to know anything for sure about the first language. Meanwhile, Old Tamil can be dated to emerge some time in the first millennium BCE. However, Old Tamil is the descendant of even older languages (it's part of the Dravidian language family, which features several languages that primarily developed in and around Southern India), and it itself evolved and changed in the millenia since: Middle Tamil reigned in the 8th-15th century CE, while Modern Tamil has been spoken for the last several centuries or so. Someone smarter than me could give an actual analysis of how the language changed, but suffice to say, between the natural and gradual passage of time, and contact with other people and other languages, there has been some shift in phonology and grammar and vocabulary that makes Modern Tamil distinct from Old Tamil.
And, well, it's not the oldest language we have evidence for, as we have evidence for even older languages, like Sumerian, which was used well over a thousand years before Old Tamil emerged, from a different language family than Dravidian.
As a point of reference, I might direct you to this older write-up of mine about the evolutions of English and Latin, as well as the links within. How does one date the beginning of English: 15th century, when the modern variant that we speak developed, or 5th century, when the first thing that we might call 'English' emerged, even if it doesn't really resemble the current language? Meanwhile, at what point does Latin stop being Latin and start being Spanish or Italian or a less famous offshoot? Languages aren't static entities that are born and die: they constantly change throughout their lifetime into multiple variations (dialects), and eventually some of those variants will differ so greatly from the others and the original that they evolve into a new language. Sometimes we can pinpoint a specific event or movement that brought about this change, but that only indicates when new developments started being introduced, and doesn't exactly tell us when the dialect actually hit the breaking point and started a new branch in the family.
The question of 'What's the oldest language?', even 'What's the oldest language that's still in use?', isn't a question that can be easily or meaningfully answered.
(It is at this point that we start to progressively stray further from my wheelhouse, as Tamil history is not my specialty—a decent amount of this is researched ad hoc, so I'm gleaning a fair bit.)
Despite this, a lot of people claim to know what the oldest language is. While they have their 'evidence', the real reason is usually pseudoscience or nationalism, rather than actual history.
Over the centuries, there has been a lot of discussion over what the 'first' language was, if such a thing could be identified. The theory for many medieval scholars was that some divinely perfect 'Adamic' language was spoken in the Garden of Eden, but that language was lost in the confusion of tongues after humans constructed the Tower of Babel (and/or, when Noah's sons spread across the continents after the Flood, they generated new nations with new languages). You may also be familiar with the legend of a king believing that babies are born knowing the 'correct' language and that others are just a corruption of it, so said king placed some newborns in a room without contact from adults to see what the natural language was. There's also the scientific monogenesis hypothesis, which I talked about in one of the answers I linked to earlier. Theories and legends abound, because people really want to know.
And for many peoples, it wasn't just interesting, it was important to them. They wanted their language to be the oldest, because that would ^somehow mean it's the best. So nations or cultures that have existed—vaguely, in some sense or capacity—for a really long time sought to claim that honor. People posited Hebrew, Chinese, Greek, Egyptian, Sanskrit, and others. And while none of those are correct, you can at least give people subscribing to that theory the most minor amounts of credit, on the basis that, at the very least, early versions of those languages are in fact very very old (if anyone suggested English or French, we'd really have a problem).
You may have also heard of the Sun Language Theory, posed by Turkish nationalists in the 1930s, which argues that all human languages are descended from a Turkic language, and again because they claim the oldest language, Turks are superior. Or, when the Indo-European language theory—the understanding that Germanic, Romance, Aryan, and other language families are all related to each other—started to really pick up steam in the 18th and 19th centuries, people in both the West and India latched on to the idea that their people date back thousands of years to a culture that was clearly influential worldwide, and is therefore… well, superior. Yes, this is getting repetitive (though it'd be remiss to ignore how this idealization of the Indo-Europeans led to Hitler's views on the Aryan race, and in turn, the Holocaust).
Tamil also gets floated as the oldest language for these same nationalistic reasons. In the early 20th century, a "Pure Tamil Movement" (தனித் தமிழ் இயக்கம், or Thani Tamil Iyakkam) developed, seeking to rid the Tamil language of influences from Sanskrit and other foreign languages. While the movement seems to have been founded by Maraimalai Adigal in 1916 and led with help by Perunchithiranar, the idea that Tamil is the oldest language was posed by Devaneya Pavanar with his 1966 book Primary Classical language of the World.
Pavanar argues that Tamil is the language from which all others are derived, and unlike other languages no word in it is arbitrary and instead it is perfect, and that it originated in Lemuria 50,000+ years ago. Lemuria, you may be aware, is a hypothetical continent from which humanity supposedly originated but it then sunk to the bottom of the Indian Ocean. One could go into further detail on this theory, but suffice to say… it is rejected by academics across the board. Among several other (way more prominent) problems with Pavanar's theory, there is no way one could realistically trace Tamil—or any other language—that far back, so there is no legitimate basis for these claims. It is all very pseudoscientific.
And yet, they persist. I can't say much about the publication history of Primary Classical language, but nevertheless this claim (along with claims that it's actually Sanskrit, and occasionally some other wrong answers) shows up on plenty of websites' "X Oldest Languages" articles, several Quora answers and YouTube or Tik Tok videos insisting on it, and most importantly, a lot of people on /r/badlinguistics complaining about the prominence of the theory.
To their credit, the Tamil language family is old. If we can call Old, Middle, and Modern Tamil one language—making it more analogous to the metamorphosis of English, than the evolution of Latin into French and Spanish and Italian and more—then over 2000 years is in fact a long time for a language to still be in use. Buuuut then we'd have to account for other languages that have also had a pokémon-like chain of development from one phase to another in that same period. If Old Tamil to Modern Tamil is a continuous stream, then how about, say, proto-Germanic to Old Germanic to contemporary German, which spans a fairly similar timeframe? I couldn't answer that question if I wanted to.
So, bringing it back to the beginning: is there proof that Tamil is the oldest language in the world?
No. No there is not.
Based on what linguists have reconstructed, the proto-Dravidian language, from which Tamil's language family developed, probably emerged in the fourth millenium BCE. Proto–Indo-European, which gave us Sanskrit and Latin and several other prominent languages and language families, was likely a little earlier than that. Proto–Afro-Asiatic emerged even earlier; from it we eventually got the Hebrew and Egyptian languages, but not for thousands of years.
The rabbit-hole likely goes on longer than that.