What problem does distilled water in a patient's bloodstream create?

Hemolysis.

Blood is isotonic with normal saline (0.9% sodium chloride). If a patient were infused with distilled water, which is hypotonic to constituent cellular elements, the cells would take up free water in an obligate fashion in order to re-establish equilibrium of tonicity across the cell membranes (intracellular vs. extracellular). This would eventually result in cell swelling and rupture.

Parenthetically, release of red blood cell contents, which is highly enriched in hemoglobin, into the plasma results in scavanging of nitric oxide (NO), which in turn results in vascular constriction resulting in decreased blood flow = ischemia (poor blood perfusion).

Patient death has been reported with hemolysis of as little as a few cc's of red blood cells (acute intravascular hemolysis).