Well Heyerdahl wasn't really arguing that the Polynesians came from South America; but that there was a South American (Peruvian) influence or cultural complex in the Pacific that predated the arrival of the Polynesians. Heyerdahl imagined that Polynesians were more likely descended from Native Americans towards the North- or some other indigenous people in Northernly America/Asia than they were descended from a common ancestor with Melanesians, Micronesians, or Indonesians.

Heyerdahl was in part buying into the Grand Migration theories of men like Sir Peter Buck who also imagined a circuitous route for Polynesians whereby they came into the Pacific from a Northernly path that took them from Asia across the Pacific to the Americans and then down and back out across the Pacific riding the currents and winds from the Northwestern American coast. Both scholars essentially argued that Polynesians drifted out into the Pacific; eventually picking up the prototypical Polynesian foodstuffs and agricultural products and the outrigger canoe in Fiji from the Melanesians population already settled there. The Polynesians, in their travels into Pacific, discovered and interacted with, largely absorbing the Andean culture that had already drifted out into the Pacific and settled the main Polynesian islands; with the major exception of New Zealand.

The problem with Heyerdahl's theory (and Buck's) and one he went at pains to explain was why Polynesian languages shared so many words with the Austronesian speakers across the Pacific, South East Asia, and even over in Madagascar, if they were in fact not related. Nor could he easily explain why the Polynesians used crops originating from South East Asia rather than the Americas. It would seem bizarre if the Polynesian or Andean settlers gave up all of their agricultural products when they met the Fijians and left no trace of ever having planted them in the Pacific (with three exceptions I will mention shortly).

Heyerdahl's theories have largely been ignored more than they have been discredited. He made some keen observations; that it was in fact possible for Andean basal-wood rafts to make it out into the Pacific, that the Sweet Potato was likely carried by voyagers out into the Pacific. However, he made some missteps too; he believed the coconut was carried from the Americas to South East Asia; now we know it was likely carried the other direction- from Oceania to the Americas. He also never satisfactorily managed to figure out why the Polynesians spoke an Austronesian language if they were not Austronesian; or why the oldest dates of settlement in the Pacific generally go from near Oceania to remote Oceania (Nearer Asia towards the Americas). Though it is important to remember that archeology in the Pacific was very much still in its infancy after World War 2 and during the time Heyerdahl was working, so a great deal of what we know now about the Lapita Cultures of near Oceania and their gradual radiating out and into the Polynesian heartland of Fiji-Tonga-Samoa, was something he never knew about.

His primary botanical evidence has not been disproven, but the way its used has changed. Rather than arguing that the sweet potato, coconut, and a type of gourd are proof that Andean settlers made it out into the Pacific and settled islands before Polynesians; we tend to interpret this evidence as proving that Polynesians voyaged to the Americas, basically exchanging the coconut for the sweet potato (though who knows if this happened on the same voyages or different ones), and exchanging chickens- or at least Polynesian chickens were given to Peruvians and bred into their livestock. The focus on Polynesians intentionally voyaging to the Americas was another rediscovery made after much of Heyerdahl's work; the voyages of the Hokule'a in the 70's and the rise of computer simulations, have redefined how most scholars think about the settlement of the Pacific. Indeed up until the 90's there was still a great deal of debate (primarily between Ben Finney & David Lewis against Andrew Sharpe) as to whether or not Polynesians could intentionally undergo long distance voyages or just short range; with long distance settlements essentially being the result of ships being blown off course, getting lost, or exiles just drifting off into the unknown. Today, long distance, intentional voyaging is accepted by scholars as an essential fact of Polynesian settlement and the main question seems to be how frequent was it and just how far did people range- but all of this really post-dates Heyerdahl.

Earlier, I said that Heyerdahl hasn't so much been disproven as simply ignored; and this is largely true. There are still people working under the assumption of his theories- though I don't know much about them. But generally his studies were so detailed and intricate, and his arguments had a certain amount of logic to them; that no one has bothered taking the time to try and refute him step by step; he has simply been left behind by more modern studies. His big studies of Easter Island have mostly fallen by the wayside as more precise carbon dating and more recent excavations have started telling a radically different story of the settlement of the island and the construction of the great Moai.

Some great reading on this topic.

• Heyerdahl, Thor. Sea Routes to Polynesia. London: Allen & Unwin, 1968. (probably the most digestible of Heyerdahl's writings, provides great summaries of his ideas and arguments, definitely worth checking out to see what he really was arguing).

• Howe, K. R. The Quest for Origins: Who First Discovered and Settled the Pacific Islands? Honolulu: Univeristy of Hawaiʻi Press, 2003. (How actually spends some time talking about Heyerdahl and gives some explanation for why no one bothers trying to disprove him, basically sayings its not worth the effort).

• Buck, Peter. Vikings of the Pacific. Chicago: University of Chicago Press, 1959. (well known for grand migration theories about Polynesians, very racially charged- he was half Maori and so wanted to establish evidence for the generally accepted belief that Polynesians were a higher race that other Pacific peoples.)

• Kirch, Patrick Vinton. On the Road of the Winds. Berkeley: University of California Press, 2002. (Good summary- though over a decade old now- of the latest in Pacific archeology and settlement theories)

• Hunt, Terry and Carl Lipo. The Statues that Walked: Unraveling the Mystery of Easter Island. Berkeley: Counter Point Press, 2012. (the latest on East Island settlement and how the Moai were made; also good work coming out of the Bishop Museum lately).