I recently heard from someone in the paleo community that the extraction process for coconut nectar kills the coconut tree just by performing the process once. I've searched the internet some and have not found evidence of this. Anyone have any insight?
No, it doesn't. You can't get coconuts from the blossoms that are "tapped", but most coconut tree farms grow both coconuts and obtain nectar from the same trees.
I was fortunate enough to visit one of these farms -- it was actually a small native grove, where they were tapping the trees to make palm wine, but the process is exactly the same, they just don't ferment and age the syrup when they make evaporated coconut sap/sugar, or regular coconut nectar. Most of the trees had several blossom spouts "tapped" with big, round bamboo tubes, and the rest of the tree was allowed to "go to seed" to produce coconuts.
The tour guide told us that, once a tree has been tapped, it will continue to produce syrup for 20 or more years, without fail.
You can learn more here, from the Phillipines, where coconut sugar/coconut harvesting is a way of life.