In Scene 4, after Cauchon describes the threat Joan poses to the Church, Warwick lays out the feudal lords' case against her. He uses a metaphor to explain the "nominal" power of the king and the real power of the feudal lords, but his explanation contains situational irony:
Nominally we hold our lands and dignities from the king, because there must be a keystone to the arch of human society; but we hold our lands in our own hands, and defend them with our own swords and those of our own tenants.
Warwick compares the king to "a keystone to the arch of human society." A keystone lies at the top of a stone arch. Wedge-shaped, it snuggles between the rest of the stones and locks them into place so that the arch won't fall. Calling something a keystone usually emphasizes its importance, but Warwick uses the metaphor a bit differently. He admits that the king holds human society together because someone has to. Still, he suggests that the king's power to distribute land is more "nominal" than anything—it is for show, so no one gets upset and demands to know why feudal lords have land while the tenants who work for them do not. He claims that "we [the feudal lords] hold our lands in our own hands." The metaphor of the king as the keystone in the arch of human society slides into the idea that the feudal lords hold up the whole world and do not need the keystone except for show.
It is ironic, though, that he insists on the feudal lords' self-sufficiency given the rest of his sentence. He claims that "we defend [our lands] with our own swords and those of our own tenants." Warwick thinks of the tenants and their swords as an extension of his property. To Shaw's audience in the 1920s, this is obviously not the case. It is not that the lords are defending their land with their tenants' swords, but rather that the tenants are defending their lords' property with no thanks except for their continued ability to live there. Lords build their wealth, and tenants put their lives on the line. It is ironic that Warwick thinks he and the other feudal lords don't need any help given that the current system relies heavily on help from the lowborn who are really holding up the arch of society.
In Scene 4, Cauchon agrees that if he cannot get Joan to repent, he will cast her out from the Church and hand her over to Warwick for punishment. Cauchon uses a simile to describe how the Church deals with heretics; Warwick seizes on the simile and turns it into a metaphor:
CAUCHON. [...] When The Church cuts off an obstinate heretic as a dead branch from the tree of life, the heretic is handed over to the secular arm. The Church has no part in what the secular arm may see fit to do.
WARWICK. Precisely. And I shall be the secular arm in this case. Well, my lord, hand over your dead branch; and I will see that the fire is ready for it.
Cauchon describes how the Church "cuts off an obstinate heretic as a dead branch from the tree of life." The tree of life appears in the Old Testament as one of the trees in the Garden of Eden. Adam and Eve are made mortal and cast out of the garden for eating fruit from the tree of the knowledge of good and evil, and they are barred from reentering the garden so that they cannot eat from the tree of life and restore their immortality. Cauchon compares an "obstinate heretic" (someone who goes against Church teachings and will not repent) to a dead branch on the tree of life. To protect the rest of the tree from the disease of this dead limb, the Church cuts it off. At that point, the "secular arm" of government can do whatever it wants with the dead limb.
For Cauchon, who still believes that he may be able to get Joan to repent, all of this is hypothetical. He uses a simile rather than a metaphor to avoid suggesting that Joan is ready to be cast out of the Church. Only if she fails to fall in line with institutional teachings will she turn into a "dead branch." Warwick, on the other hand, is all too eager to execute Joan. He assures Cauchon that he is ready to be the "secular arm" of which Cauchon speaks. He tells Cauchon to "hand over your dead branch," as though Joan's "obstinacy" as a heretic is a foregone conclusion. The metaphor dehumanizes her. By calling her a "dead branch" instead of a human woman, and by referring to her as "it," he makes it seem only logical that he is going to throw her on a fire. Burning a human might look closer to madness than sanity, but burning a dead branch makes sense. Warwick is careful to pick up on the language Cauchon is using so that Cauchon will be able to square the mistreatment of Joan with the institutional position of the Church.
The play does not paint either man in a flattering light. Warwick is bloodthirsty and manipulative, and Cauchon is so invested in speaking for the Church that he fails to think for himself and spot how Warwick is twisting his words. Neither man considers that Joan might be a live branch, thinking for herself and growing in a new direction.
In Scene 6, Stogumber finds that he is deeply disturbed by Joan's execution, which he just witnessed. He uses imagery to describe both his experience and Joan's, drawing a metaphorical comparison between her execution and the fate of those who have condemned her:
You madden yourself with words: you damn yourself because it feels grand to throw oil on the flaming hell of your own temper. But when it is brought home to you; when you see the thing you have done; when it is blinding your eyes, stifling your nostrils, tearing your heart, then—then—[Falling on his knees]. O God, take away this sight from me! O Christ, deliver me from this fire that is consuming me! She cried to Thee in the midst of it: Jesus! Jesus! Jesus! She is in Thy bosom; and I am in hell for evermore.
Stogumber emphasizes the physical sensations he experienced while watching Joan burn. He describes how horribly different these feelings are from the "grand" feeling of "throw[ing] oil on the flaming hell of your own temper." It feels good to fan the flames of vengeance against a "heretic" or rebel like Joan. But smoke and fire in the eyes and in the nostrils, and the horror in the "heart" of anyone with human sympathy for Joan, chokes out this good feeling. Watching Joan burn has made it evident to Stogumber that to damn her to this horrible fate is also to "damn yourself" in a religious sense.
He draws this metaphor out through the rest of this speech. "O Christ, deliver me from this fire that is consuming me! She cried to Thee in the midst of it: Jesus! Jesus! Jesus!" is a deliberately unclear line. Stogumber could be reporting Joan's pleas to God while she was burning, but it seems just as likely that he is himself calling to Christ and pleading for salvation from "this fire that is consuming me." The next line clarifies that the latter is at least part of what he means. He addresses Christ directly, stating that "She [Joan] is in Thy bosom; and I am in hell for evermore." Stogumber is no longer just comparing the experience of watching the execution to the experience of being actually burned alive. Now, he is suggesting that his fate is worse: Joan burned physically but gets to spend eternity in heaven. Stogumber, on the other hand, has damned himself to eternity in the flames of hell.
In Scene 6, after Joan's execution, the executioner reports to Warwick that the job is done. He seems to be speaking literally, but his report is tinged with dramatic irony and metaphorical meaning:
THE EXECUTIONER. Her heart would not burn, my lord; but everything that was left is at the bottom of the river. You have heard the last of her.
The executioner means that the physical organ of Joan's heart refused to burn. Although organ tissue is not necessarily the part of the body that is most resistant to burning, it is true that it requires extraordinarily hot fire to completely incinerate a body. Part of the executioner's job, if burning people is a form of execution, is disposing of remnants that have not been reduced to ash. He is simply telling Warwick that he has burned Joan and disposed of the remains in the river. He genuinely believes that Warwick has "heard the last of her."
But anyone reading or watching Shaw's play in the 20th century and beyond knows that Joan will only get more famous after her death. Centuries later, people are still talking about her. The whole reason Shaw was inspired to write Saint Joan is because the Catholic Church canonized her as a saint as recently as 1920. This future knowledge allows the audience to enjoy a sense of dramatic irony at the executioner's assertion that Warwick and the world have heard the last of Joan. On the contrary, her execution has ensured that the world will never stop hearing about her. Additionally, the fact that Joan has been made a saint gives metaphorical meaning to the phrase "Her heart would not burn." Saint Joan represents extraordinary faith and commitment to her cause. Whereas the executioner thinks of her as a mere human woman, the audience knows her as a legend. Her fire-resistant "heart" becomes not just an organ, but her strength of character.