What would happen to an erythrocyte if pure water were administered intravenously?

The water would rapidly pass into the erythrocytes through their cell membrane, causing the cells to swell up and eventually rupture.

Erythrocytes cannot survive in pure water.  They would rupture.  To inject pure water intravenously would cause serious injury or death in the patient, depending on the volume of water infused.

The others are correct.  They would swell and rupture.  The next question is why?  
The answer is osmosis.  Osmosis is the diffusion of water through a semi-permeable membrane from a area of lower concentration to an area of higher concentration.  A semi-permeable membrane is a membrane that allows some things though but not others.  In this case water can pass through the membrane of the erythrocyte(think of this as the 'skin' of the erythrocyte).  Osmosis is concentration driven.  If the fluid you add to the vein is water then concentration outside the erythrocytes will be much less than the fluid inside the erythrocyte. Via the process of osmosis water will diffuse from the vein into the more concentrated erythrocyte.  This extra fluid cause swelling and may eventually lead to rupture and erythrocyte death.