It is possible to run out of the tears. However that could be so dangerous for the health of the eyes which will cause the blindness. The tear glands system will keep the balance of the water of the eyes. If you cry all the time, the serious dryness of the eyes will cause a lot of deadly eyes problems which you need to pay attention to.
I think it is just a metaphor used in novels to express how hard the people cried. Tear is produced by glands in eyelids to lubricate, nourish and protect eyes. In normal cases, people won't running out of tears, because dozens of glands in eyelids are producing the tear. But excessive evaporation or drainage of tear would give you a dryness of eyes. In some other extreme cases, if people are dehydrated, people would decrease sweeting and urination. In this case, people might run out of tears becaused they would gradually stop tear producing for dehydration. It could dangerous if people stop to produce tear. Tear is very important in eyes. Without it as lubrication, we might feel miserable even to rotate eyeballs, not mention any other movements. Intake enough water is essential to meet the needs of water consumption everyday.