Indeed, the white dominated government of then Rhodesia officially established that Great Zimbabwe could not have been built by an indigenous African civilization. In a spectacular show of circular reasoning, they concluded that African people were not capable of such advanced architecture and therefore, any remains found must have been built by prehistoric colonists. Specifically, they speculated that Zimbabwe could have been built by descendants of the biblical Queen of Saba Sheba (who was of course assumed to have been white). As far of destruction of evidence, the procedure was a bit more oblique: government awarded licenses to dig in the site primarily as a treasure hunting endeavour, closer to mining than to archaeology. Eventually, of course, the apartheid regime of Rhodesia fell and the new, nationalist government changed the name of the country to Zimbabwe, drawing inspiration and pride from the archaeological site and the civilization that built it (which is exactly what the old regime had tried to avoid).
Source: Bruce G. Trigger, A History of Archaeological Thought
EDIT: I must say I'm on mobile now, I will probably revise this answer a bit later.
There is no truth to the theory that Great Zimbabwe was built by a non-black people. There is evidence of poor archaeological practices that hindered further research done on the site. Modern archaeological work has uncovered more artifacts. (Edited for clarity)
Early excavations sponsored by Rhodes were for the purpose of supporting an imperial narrative that white settlers had created. The narrative was that indigenous African peoples were primitive, and these primitive people were not capable of the change that Great Zimbabwe clearly embodied. So, the idea was that the stimulus for change could only come from outside influences (in this case the outside influences were of Mediterranean origin). Rhodes himself was convinced that the site was constructed by Phoenicians and other early archaeologists also speculated on the site being occupied by Egyptians and other archaic civilizations. Early work done on the site by men like J. Theodore Brent uncovered artifacts that we would now consider to be of very clear African origin. However Brent, having spent quite a bit of time studying the Phoenicians prior to his arrival in South Africa, chose to link the artifacts he found at Great Zimbabwe to the Phoenicians noting that, "a prehistoric race built the ruins ... a northern race coming from Arabia ... closely akin to the Phoenician and Egyptian." (1)
Following Brent's fieldwork, Richard Nicklin Hall wanted to prove Brent's findings and firmly establish the origins of Great Zimbabwe as being of non-black descent. Hall tore apart the site by excavating large amounts of stratified deposits all at once. (2)
As /u/WillieJGrafton, there is plenty of archeological evidence that would support the existence of early peoples occupying the site.
Sources:
Mysteries of Great Zimbabwe over at PBS: http://www.pbs.org/wgbh/nova/ancient/mysteries-of-great-zimbabwe.html
The Burden of Tribalism by Martin Hall: http://www.jstor.org/stable/280354
Note: This is my first r/AskHistorians post, so any criticism regarding format or substance is entirely welcome. As a student, I'm always looking to improve.
Anyone has a source concerning the destruction of evidence in particular? I've heard that Rhodesia only allowed researchers that beforehand agreed that Great Zimbabwe wasn't locally-made (deep into the 70's), but I've never heard of actual destruction of material.