The Blessed Palms that we are given on Palm Sunday are sacramentals. According to the Catechism of the Catholic Church, sacramentals are “sacred signs which bear a resemblance to the sacraments. They signify effects, particularly of a spiritual nature, which are obtained through the intercession of the Church.” Other sacramentals include relics, holy oils, the rosary. Palms are blessed and distributed to the faithful during the celebration of the Passion of the Lord, on Palm Sunday, commemorating the multitude bearing palms who triumphantly led Christ in procession on that day in Jerusalem.
The tradition of blessing palm can be found in writings as far back as the time of Saint Bede the Venerable, 673-735. The liturgy of Palm Sunday, or Passion Sunday, refers simply to branches. In the prayer of blessing, the priest says, "Almighty ever-living God, sanctify these branches with your blessing, that we, who follow Christ the King in exultation, may reach the eternal Jerusalem through Him. Who lives and reigns for ever and ever." During the procession with the palms the people walk in joyous public demonstration of their loyalty to Christ.
Given that Blessed Palms are sacramental, the Church teaches that they can be disposed of in two ways. First, some parishes will announce, usually just after the beginning of the new calendar year, that they are looking for old palms to be turned in. These parishes then burn the old palm to make the ashes used for the upcoming Ash Wednesday services. The second way to dispose of old palms is to burn the palm yourself and return the ashes to the ground, perhaps by spreading them in a garden.