Cremation was a pagan practice dating back to the Iron Age and open air pyre is an ancient Hindu tradition still practiced in India, Nepal and Eastern religions.
Christians who believed in the physical resurrection of the body, did not approve of cremation. The body of Jesus himself, whose example Christians follow, was ceremonially wrapped and placed in a tomb. He was resurrected three days later, with the piercings from the stakes in his hands still visible.
While the Bible doesn't specifically forbid cremation, it does explain that the custom of the Jews was to bury their dead (John 19:40).
The apostle Paul, stated that the earthly bodies of the dead are sown (planted) and raised in a spiritual body (1 Corinthians 42-44) to either everlasting glory or eternal judgement. Paul also taught that the body is the temple of the Holy Ghost (1 Corinthians 3:16-18).
The incineration of corpses in the Bible was almost always a curse or punishment. The Bible also speaks against those who pass their children through the fire (Leviticus 18:21). These teachings are some of the reasons many devout Christians have rejected cremation.
In fact, the practice of cremating the bodies of the deceased was outlawed and made punishable by death in the year 789 by Charlemagne, also known as Charles the Great and Holy Roman Emperor from 800 to 814 (crowned by Pope Leo III). It wasn't until a newly invented cremating device was introduced at the Vienna Exposition in 1873, that cremation became a topic of debate again. Queen Victoria embraced the idea, but the church fiercely opposed it and would not allow the burial of cremated ashes on consecrated ground. Three years later, in 1876, a man by the name of Dr. Francis Julius LeMoyne, also inspired by the model presented at the Vienna Exposition, built the first crematorium in America. It was built on his property in Pennsylvania, sparking much controversy.
In 1884, a Welsh man named Dr. William Price carried out the first cremation in the United Kingdom in modern times. In what appears to have been a fit of anger and defiance against God, he named his stillborn son "Iesu Grist", the name of Jesus Christ in the Welsh language. When he attempted to cremate the infant's body, the townspeople intervened and he was arrested. He was later acquitted and carried out the cremation ceremony, complete with pagan prayers, on March 14, 1884.
Parliament passed the Cremation Act in 1902 and Mormon, Baptist, Protestant, and Catholic churches slowly began allowing the practice, although discouraging it. By the 1960s it was becoming widely accepted.
Today, cremation has become much more widely accepted with about half of dying Americans opting to be cremated. It is especially convenient in urban areas where land is scarce and burial plots are more costly. Only the Eastern Orthodox Church continues to forbid cremation, making few exceptions.
Sources:
[1] Bregman, Lucy (2010). Religion, Death, and Dying. 3. ABC-CLIO. p. 13. ISBN 978-0-313-35173-0.
[2] Hutton, Ronald (2009). Blood and Mistletoe: The History of the Druids in Britain. New Haven: Yale University Press. ISBN 978-0-300-14485-7.
[3] Encyclopædia Britannica: Or, A Dictionary of Arts and Science ..., Volume 9, p. 263.