I think that human beings pretty much are unable to live without hope. We certainly see that in O. Henry's portrayal of Johnsy in this story.
I believe that we human beings have to have hope to live. We need to be able to feel that tomorrow will be as good as or better than today. We need to feel that our lives can always improve. Otherwise, what is the point in being alive?
You can see the need for hope in the case of Johnsy. When she did not have hope, she was pretty much convincing herself to die. But once the old man painted the leaf, she had hope and convinced herself to live.
In O'Henry's story "The Last Leaf" hope plays the role of motivation. The girl develops pneumonia and becomes very weak. She is physically and emotionally drained after being so sick. She begins to give up. Everyday she watches a vine on a wall waiting for the last leaf to die so that she too can die and rest from her illness.
The artists has painted the leaf on the wall, but the girl does not know it. She watches for the leaf to fall and when it does not, she becomes motivated by its ability to hang on. She begins to heal after realizing that if one tiny leaf can survive, so cane she.