AP3431 burning up with no load on it
I think my problem here might have been the use of acid-based flux, intended for plumbing, rather than electronics.

From the data sheet: 


  Like other inorganic acid fluxes, Kester SP-30 is too corrosive for
  electrical or electronics soldering applications.


Oops :). I'll need to use something a little less crazy in my future projects.