Oh, crud! I forgot; the Church doesn't believe in capital punishment anymore. :-(
Actually, the Church still recognizes capital punishment as valid. (CCC art.2266). Despite the protestations of this current "conservative" Pope. OTH, should government have that much power over life and death?
Not so fast.
The Church discourages capital punishment if the particular state or nation has the resources and the facilities to keep the perp jailed for life.
Now in the case of burning a heretic, it just so happens that the Church does NOT have the facilities to keep the so-and-so jailed for life.
QED: Bonfire Time!