Christian "martyrs" were killed, not because of their faith, but because they happened to be a convenient group upon which to blame Rome's woes
"In Rome, a huge fire destroyed much of the city. Nero opened up public buildings to house the homeless, but rumors that he had been singing and dancing while Rome burned turned public opinion against him. He looked for a scapegoat and found one in the Christians.
Jesus had died 30 years earlier, but energetic missionaries, such as Paul, had spread his message across the empire. Rome had relatively few Christians and they were not widely trusted. Nero rounded them up and executed them brutally, throwing some to the lions, burning others and crucifying many more. "
PBS Special: The Roman Empire in the First Century
So, it wasn't because of their refusing to renounce their religion that the Christians died, but because they were in the wrong place at the wrong time.
"The founder of that sect, Christ, had been executed. His death had briefly suppressed the destructive cult, but again it erupted, not only in Judea, the birthplace of the evil, but also in Rome where shameful atrocities fester and spread."
Karen King, Professor of New Testament Studies and the History of Ancient Christianity, Harvard University:
The Christians would have been a good target. After all, their main hero was a criminal who had been put to death by Roman order. In addition to this, they were doing things like exchanging a kiss among brothers and sisters at their meetings, which sounded a little bit like incest. They were also eating the body and drinking the blood of their God, which sounded a bit like cannibalism.
Nero rounded up all the Christians in the city. They were hideously tortured and executed. Then Nero plundered the empire for funds. Temples were robbed of their statues. Treasures that generations had dedicated to the greatness of Rome were absorbed into imperial coffers.