She will take him to visit her church and meet her church community there. He will burn up - it's difficult to say what he means - it may be an Americanism, but it could mean that the experience would move him deeply.

"I'm gonna take him to my church" It is important to note that she says "MY church". You will also hear people say "go to church" without a definite article or possessive pronoun, but it would mean something different if she said "I'm gonna take him to church". Without 'my', it just means she will take him to any church, maybe it is one she goes to all the time, maybe not, we don't know. The fact she says "I'm gonna take him to MY church" tells us this is the church she goes to all the time and has a connection with. She doesn't own the church. "I'll burn up" It's hard to know exactly without context, but people who are not Christian or would be considered 'sinful' by Christians sometimes joke about 'burning' if they come into contat with Christian things, e.g. touching a Bible, walking into a church, having holy water thrown on them, etc. So I would guess that he is not a Christian and probably lives in a non-Christian way and is joking that he would 'burn up' (be on fire) if he walked into a church.