What are the biochemical ramifications of ingesting of distilled water?
As @DavePhD has noted, drinking small amounts of distilled water does not pose any significant adverse health consequences.  However, the World Health Organization (WHO) has studied the effects of long-term drinking of distilled water and issued a report.  They investigated the following five areas of potential adverse human health impact:


Direct effects on the intestinal mucous membrane, metabolism and mineral homeostasis or other body functions.
Little or no intake of calcium and magnesium from low-mineral water.
Low intake of other essential elements and microelements.
Loss of calcium, magnesium and other essential elements in prepared food.
Possible increased dietary intake of toxic metals.


In their conclusion, they state,


  "Drinking water should contain minimum levels of certain essential
  minerals (and other components such as carbonates). Unfortunately,
  over the two past decades, little research attention has been given to
  the beneficial or protective effects of drinking water substances."


and


  "Demineralised water that has not been remineralized, or low-mineral
  content water – in the light of the absence or substantial lack of
  essential minerals in it – is not considered ideal drinking water."


(emphasis mine)