No. Catholics venerate relics; they do not worship them. Worship, or adoration, is given to God alone; veneration is the respect and love shown to a holy person, and it passes through the saint to Christ. Catholics honor relics because the saints' bodies were, in the words of the Holy See's 2017 Instruction, living temples of the Holy Spirit and are destined for the resurrection. The practice goes back to the second-century Church, which gathered the bones of the martyr St. Polycarp.
The distinction has been stated in every age. The account of Polycarp's martyrdom says Christians worship Christ and love the martyrs as his disciples (quoted in Catechism 957). St. Jerome wrote that we do not worship or adore relics, lest we bow to the creature rather than the Creator. The Council of Trent (1563) and the Second Vatican Council (Sacrosanctum Concilium 111) both affirmed the veneration of authentic relics.
The Church also admits the abuses: forged and traded relics were denounced as early as St. Augustine and helped provoke the Reformation's rejection of the practice. Canon 1190 states that it is absolutely forbidden to sell sacred relics, and the 2017 Instruction asks that relics be honored without superstition. The veneration of relics is a form of popular piety, which the Church approves and regulates; it is not an obligation laid on every Catholic.
Related questions, answered
Is venerating relics in the Bible?
The Bible does not command it, but it records the pattern. A dead man revives on touching the bones of the prophet Elisha (2 Kings 13:20-21); a woman is healed by touching the hem of Jesus' cloak (Matthew 9:20-22); and handkerchiefs that had touched St. Paul are carried to the sick (Acts 19:11-12). In each case the power is God's, working through a person close to him.
Isn't it morbid to keep bones and bodies in churches?
It can feel that way at first, and the reaction is common among converts and cradle Catholics alike. The practice makes sense from the Christian belief in the resurrection of the body: these remains are treated with honor because they will rise, and because the person they belong to is alive with God. The earliest account of a martyr's relics, that of St. Polycarp, already has Christians gathering at the grave to keep the anniversary of his death.
Do Catholics have to venerate relics?
No. Veneration of relics is a form of popular piety that the Church approves and regulates, not an obligation. What the Church asks every Catholic to hold is the reality behind it: the communion of saints and the resurrection of the body.
