Are there any distinct biochemical reactions that are affected by the ingestion of distilled water?
The water is "distilled" in its literal sense.
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:
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."
"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."