It does not literally mean executed, at least not in most contexts. It also does not literally mean “burn”, at least in most contexts. Instead, “I would see him burn” means “I would rather that something terrible happen to him”. So here, the sentence means “I care so much about her that I would do/allow anything to protect her, including causing/allowing something terrible to happen to him.”

Seeing him ''Burn'' in this context is an expression. Describing the way or the level of protectiveness this person is willing to have towards the person in question which in this case is the female.

In this context it means. The person would not care or bother if another person, who they believed was destroying someone they cared about. Was to burn (burn is not used in a literal sense). "I will not let him destroy her, I would watch him suffer great pain, if it he did not cease destroying her" a parent would say this about a man who was being abusive to their daughter. I do not know but I suspect it may come from the Public burning of Catholics in England. You can not and must not shy away from the historic evolvement of a nations language that was, and still is evolving due to historic events. Each new historic phase produces new expressions that learners will always enquiry about or puzzle over. yes you could swap executed with burn.