Why is the sound of fingernails on a chalkboard so intolerable?
My initial thoughts are that the sound produced a dissonant sound, that is (from the link):


  "An unstable tone combination is a dissonance; its tension demands an onward motion to a stable chord. Thus dissonant chords are 'active'; traditionally they have been considered harsh and have expressed pain, grief, and conflict."


—Roger Kamien (2008)

According to the article "Psychacoustics of a chilling sound", (Halpern et al. 1986), it is the low frequency components of the sound that cause the most discomfort. In their tests, removing the low frequency part in a digital simulation, made the sound a bit more bearable.

In the conclusion of the article, there is some support for the Wikipedia article that Josh Gitlin linked to in his question:


  the complex acoustic stimulus pictured in Figure I
  very closely resembles some of the spectrograms of warning
  cries emitted by macaque monkeys (Green, 1975). As
  another possibility, the signal may be similar to the vocalizations
  of some predator.