The Queen's Relationship to the Stag (Dramatic Irony)
While the queen is very honest with those in her closest circle, she keeps her feelings about the stag and its death hidden. But we, the viewers, are able to see her real feelings. She asks Philip about his hunting as though she is hopeful about his being able to kill the stag, but then when she sees the stag in nature, she urges it to run away to avoid getting killed, a smile spreading across her face when the stag manages to escape death for the moment. Then, when the stag has indeed been killed and she views the body, she speaks to the proprietor of the hunting lodge as though she's happy to see it dead, but privately she looks very upset. Viewers are able to see this, thus creating a disparity between what we know and what the characters know—an instance of dramatic irony.
Tony and Cherie's Conversations (Dramatic Irony)
A tension that is present from the start of the film concerns the fact that Tony Blair and his wife have anti-royalist sympathies and he won the election based on a progressive campaign to modernize Britain. Thus, his conversations with his wife, especially before their first meeting with the queen, are not exactly complimentary towards the royal family. They disparage the royal family, especially Cherie, and do not have very much respect for the monarchy. In the sense that we the viewers are privy to the Blairs' private opinions about the royals, this is an example of dramatic irony.
Tony Defends the Queen (Situational Irony)
Tony Blair begins the film as a progressive prime minister who does not care for the monarchy. Furthermore, the royals make his life harder when they refuse to bend to public opinion surrounding an appropriate response to Diana's death. In spite of this, however, Tony's dealings with the queen only make him sympathize with her more. By the end of the film, he has decided that he has respect for the monarch and believes that she did her best given the circumstances. This is a rather ironic reversal for an anti-royalist.
The Queen Bends to Public Expectation (Situational Irony)
The fact that the queen bends to citizens' expectations of her, breaking royal protocol and coming back to fulfill all of Tony's expectations for her, is an example of situational irony. The queen is defined by her sense of duty and her belief in tradition, yet she bends to it when she realizes that her popularity is plummeting. Even though her public grief is somewhat disingenuous, merely a way to pacify the public, she does make the effort, in order to keep judgment at bay.