1
general::
Execute someone by tying to a stake and burning; also, punish severely. This expression refers to a method used in the Middle Ages for putting heretics to death, but now it is used as a hyperbolic metaphor for harsh punishment, as in She was sure she'd be burned at the stake for losing the contract. In fact, the stake can be used loosely for any extreme punishment. William Makepeace Thackeray so used it in Henry Esmond (1852): "I know I would go to the stake for you,' said Harry."
American Heritage Idioms
2
general::
Phrase(s): burn someone at the stake
1. Lit. to set fire to a person tied to a post (as a form of execution). • They used to burn witches at the stake. 2. Fig. to chastise or denounce someone severely or excessively. • Stop yelling. I made a simple mistake, and you’re burning me at the stake for it. • Sally only spilled her milk. There is no need to shout. Don’t burn her at the stake for it.
McGrawhill's American Idioms And Phrasal Verbs