Reference:
This has been going on for a while sometimes in the name of christianity
But putting someone to death is not in keeping with christianity as it should be practiced IMO.
It all comes down to that infamous verse in Leviticus chapter 20 verse 13 where such people "shall surely be put to death". But just before then verse 10 makes the same requirement for adulterers. Leviticus is part of the Old Testament which forms the basis for the Jewish faith.
But after the Old Testament was written, Jesus came along, and although Jewish was a bit of a rebel and many of his teachings rewrote the rulebook and eventually the New Testament was created based on those teachings. And Christians should follow the New Testament in preference to the Old Testament. The Gospels in the New Testament form the core of Jesus' teachings, so what do they say about homosexuality? Nothing.
But Jesus does say something about adultery. From John chapter 8 (in the King James version):
3And the scribes and Pharisees brought unto him a woman taken in adultery; and when they had set her in the midst,
4 They say unto him, Master, this woman was taken in adultery, in the very act.
5 Now Moses in the law commanded us, that such should be stoned: but what sayest thou?
6 This they said, tempting him, that they might have to accuse him. But Jesus stooped down, and with his finger wrote on the ground, as though he heard them not.
7 So when they continued asking him, he lifted up himself, and said unto them, He that is without sin among you, let him first cast a stone at her.
8 And again he stooped down, and wrote on the ground.
9 And they who heard it, being convicted by their own conscience, went out one by one, beginning at the eldest, even unto the last: and Jesus was left alone, and the woman standing in the midst.
10 When Jesus had lifted up himself, and saw none but the woman, he said unto her, Woman, where are those thine accusers? hath no man condemned thee?
11 She said, No man, Lord. And Jesus said unto her, Neither do I condemn thee: go, and sin no more.
And I have no reason to doubt that Jesus would have said something similar if it had been a homosexual, though the question of sin is one which only God can confirm one way or the other.