Julien Riel-Salvatore reviews some reasons why kuru did not wipe out the Neandertals.
I don't have anything to add. The hypothesis comes from a paper in Medical Hypotheses by Simon Underdown; here's part of the abstract:
TSEs could have infected Neanderthal groups as a result of general cannibalistic activity and brain tissue consumption in particular. Further infection could then have taken place through continued cannibalistic activity or via shared used of infected stone tools. A modern human hunter-gatherer proxy has been developed and applied as a hypothetical model to the Neanderthals. This hypothesis suggests that the impact of TSEs on the Neanderthals could have been dramatic and have played a large part in contributing to the processes of Neanderthal extinction.
The short paper is admittedly speculative but quite clear. It does fail to cite the literature about selection on the prion gene, PRNP (I discussed it here in early 2006).
Riel-Salvatore points out all the reasons why it is probably wrong:
1. Neandertals were eating each other 100,000 years before they went away.
2. Neandertals didn't live as long as most humans who develop TSE symptoms.
3. Neandertals lived at much lower densities than the Kuru-spreading Fore people, and it's not credible for them to have spread a prion disease by cannibalism across this space (although, the urine-dispersed CWD seems to do spread pretty well through deer).
4. Non-Neandertals have a clear record of altering human skeletal remains also, including African Middle Pleistocene and early Upper Paleolithic Europeans.
I think these points are fatal to the hypothesis, unless we resort to a different mode of transmission; but in that case there is no reason to suppose that a prion disease would be involved rather than a viral or bacterial agent. I should also mention that despite early claims, there is not any reason now to think that the human prion gene was under long balancing selection.
Underdown S. 2008. A potential role for Transmissible Spongiform Encephalopathies in Neanderthal extinction. Med Hypotheses (in press) doi:10.1016/j.mehy.2007.12.014