Which parts of chapter 5 in "The Scarlet Pimpernel" demonstrate irony?

The two most common types of irony are situational irony and dramatic irony. In situational irony, both the character/characters and audience are unaware of the true situation: the outcome of events do not match expectations. Here are examples of situational irony.

Dramatic irony occurs when the audience knows facts about the character (or specific clues of the story) that the character is not privy to. (Please refer to the link for examples of dramatic irony).

In Chapter 5 of The Scarlet Pimpernel, a good example of dramatic irony is the moment the Comtesse forbids her daughter, Suzanne, to speak to Lady Marguerite Blakeney.

"Suzanne, I forbid you to speak to that woman," said the Comtesse, sternly...She had spoken in English so that all might hear and understand;... As for Lord Antony and Sir Andrew Ffoulkes, their very hearts seemed to stand still with horror at this gratuitous insult... instinctively both glanced hurriedly towards the door, whence a slow, drawly, not unpleasant voice had already been heard.

"We are in England, now Madame," rejoined the Comtesse coldly, "and I am at liberty to forbid my daughter to touch your hand in friendship. Come, Suzanne."

The dramatic irony is that the Comtesse and all her family owe their liberty to the Scarlet Pimpernel, who is the alter-ego of Sir Percy Blakeney, Lady Blakeney's husband. In the story, it is the Scarlet Pimpernel who continues to be responsible for the daring escapades of various aristocratic families from the Guillotine. As such, the Comtesses' insult is ironic: she does not know that she is insulting the wife of the man responsible for her own rescue.

Another irony is that Lady Blakeney herself does not yet know her own husband is the Scarlet Pimpernel, the man (along with his courageous band of followers) responsible for rescuing the Comtesse's family from the blood-thirsty clutches of Madame La Guillotine.

What is an example of irony in The Scarlet Pimpernel?

An example of irony in The Scarlet Pimpernel is that Marguerite confides in her husband that her brother is in trouble for being accused of assisting the mysterious Pimpernel, and he turns out to the be the Pimpernel.

Irony is when something happens that is the opposite of what is expected.

The baronet Sir Percy Blakeney has been separated from his wife Marguerite ever since she caused the execution of the sons of the Marquis.  When her brother Armand is in danger, Marguerite decides to go to Percy for help.  He finds her request for help ironic.

[The] murderous dog of the revolution is turning upon the very hands that fed it? . . . (ch 16)

Although Percy blames his wife for her role in the Marquis’s sons deaths, he realizes that he needs to protect Armand.  After all, Armand has been accused of being in the League of the Scarlet Pimpernel.

When Marguerite finds the ring in her office, she realizes that her husband is the Scarlet Pimpernel.

Oh! how could she have been so blind? She understood it all now—all at once . . . that part he played—the mask he wore . . . in order to throw dust in everybody's eyes. (ch 19)

Now Marguerite realizes that she should have known all along.  She is concerned about her brother’s safety and her husband’s.

The Scarlet Pimpernel by Baroness Orczy is set during the French Revolution. During this time, of course, the people were bloodthirsty for the heads of aristocrats and were desperate to punish anyone who was perceived as being (and sometimes actually were) responsible for their poverty and despair. 

As the story opens, we are introduced to the world of deception, violence, and intrigue in the middle of a revolution. At the gates in and out of Paris, carts and carriages are routinely searched for aristocrats who are trying to escape certain beheading. More and more of the aristocrats who are in prison are managing to escape, primarily through the efforts of a mysterious man who calls himself the Scarlet Pimpernel.

One particular guard, Sergeant Bibot, is exceptionally skilled at ferreting out these hidden aristos, and everyone has begun to gather at his gate to watch the spectacle of their hated enemies, the aristocrats, being discovered and sent back to prison. Bibot routinely makes a dramatic spectacle of these moments.

One day Bibot tells everyone the story of a fellow guard who foolishly let some aristocrats get through. Even worse, the driver of the cart was the notorious Scarlet Pimpernel, which made the hapless guard a target for everyone's mockery. Bibot is merciless in his mockery of the poor guard. In fact, partway through the story,

Bibot was laughing so much at his own tale that it was some time before he could continue.

Soon, an old hag drives a cart to this gate. Bibot is disgusted by the woman, as she collects the hair of the beheaded aristocrats. When she announces that she is going to take care of a sick child--perhaps suffering from smallpox--Bibot has no interest in detaining her (for fear of being contaminated by the dread disease) and lets her pass through the gate.

Minutes after the woman is gone, an officer rides up to the gate and asks if an old woman has tried to pass through the the gate--an old woman who will probably claim that she is taking care of someone with smallpox. Bibot is horrified when he learns that there were aristocrats hiding in the cart; even worse, he is mortified to learn that the old woman was the Scarlet Pimpernel in disguise.

Dramatic irony occurs when there is a contrast between what is expected and what happens, a scenario which generally promotes ironic humor. The irony of this situation, of course, is that the very thing Bibot so arrogantly and mockingly taunted the other guard for doing is the same thing that just happened to him. It is a perfect irony.