And, behold, a woman in the city, which was a sinner, when she knew that Jesus sat at meat in the Pharisee's house, brought an alabaster box of ointment, And stood at his feet behind him weeping, and began to wash his feet with tears, and did wipe them with the hairs of her head, and kissed his feet, and anointed them with the ointment..
In the Bible lands and times people would save their righteous tears in a bottle to prove to God that they had wept many times for a Godly cause. Tears of anger and other "unrighteous" tears were not saved. Only tears cried for the glory of God were kept. Tear bottles were buried with a person when they died to show how many times they had cried for the glory of God. The woman in Luke 7 did not cry so much that she could wash Jesus' feet (although she was weeping). She brought her tear bottle and washed his feet with her tears to show her love. This was a great sacrifice as she would not have the tear bottle when she died. The verse in Psalm 56 is saying that God's people do not need to save their tears, since things done for the glory of God are kept on record in "God's book" by God Himself.