Globally: There is debate
On the science front, you can find arguments that either the decline in CO$_2$ during the Eocene-Oligocene transition (about 34 million years ago) or the opening of the Drake passage (anywhere from 17-41 million years ago, depending on the source) is the primary culprit. Probably, both were involved.
There are a lot of interlocking theories about the connections between CO$_2$ levels, ice ages, and the isolation of Antarctica by the circumpolar current that are, at best, not fully explained. From a scientific point of view, it is probably best to say that we don't realy know what the effect on world climate would be.
Locally: No change
The Humboldt current that dominates Chilean climate is caused by the 'point' of South America blocking cold circumpolar water and driving it north. Along the coast of Chile it causes upwelling currents which leads to both rich marine productivity, and a specific sequence of coastal climactic zones: temperate rain forest, Mediterranean, and desert.
The climactic sequence is seen on the West coast of all continents. In South America, there are the Valdivian temperate rain forests, the Mediterranean region around Santiago and Valparaiso, and then the Atacama desert. A very similar combination is seen in North America, with the temperate rain forests of the Pacific Northwest, the Mediterranean climate of California, and then the deserts of Baja California. The west coast of Africa doesn't extend far enough south to get the temperate rain forests, but there is the Mediterranean region around Cape Town folowed by the Namibian desert. Europe is a bit different, because of the geometry of the continent and the Gulf Stream, but you can still see a similar pattern. Ireland, Brittany, and Galicia Spain have a climate similar to a temperate rainforest, while the actual Mediterranean is Mediterranean and the Sahara is the desert.
Since this pattern is general, whether an Antartic circumpolar is being deflected towards the equator or not, it is safe to assume that the climate in Chile won't be changed too much by the closure of the Drake Passage...that is unless that closure causes the ice age we are in to end or something drastic.