Royal isn't actually ill (Dramatic Irony)
The greatest deception in the film is the fact that Royal lies to his family and tells them that he is terminally ill with stomach cancer, so that they will invite him back into his old house and take care of him. He enlists the help of Pagoda, and gets a loyal hotel worker to pose as a doctor to give his claims some credence. All the while, the viewer knows that Royal is lying and is not actually ill. Thus, there is a dramatic irony in the fact that the viewer knows that Royal is grossly deceiving his family, even as his family members have complex emotional responses to the news of his illness. The disconnect between what we the viewers know and what the characters know creates dramatic irony.
Royal ends up dying (Situational Irony)
Soon after Royal has redeemed himself in the eyes of his family and made amends for the fact that he lied about being on the brink of death, he actually dies of a heart attack. This creates for an instance of situational irony. While the big revelation was previously that Royal wasn't actually dying and was simply lying in hopes of being redeemed in his family history, he dies not longer after everything has been resolved. Thus, after spending the film resenting Royal for staging his own death, the Tenenbaums must mourn and come to terms with a completely non-artificial and untimely death.
The Title (Situational Irony)
The title of the film suggests that the Tenenbaums themselves are a kind of "royal" family, an enviable and illustrious fixture in their community. In fact, they are a fallen dynasty, a shell of a happy and prosperous family, and their patriarch "Royal" is far crasser than the king one might expect to be at the helm of such a gifted group. The irony lies in the fact that while we expect a family labeled "royal" to be special and elevated, the Tenenbaums are anything but; in fact, they are quite a mess.
Margot's affair with Eli (Dramatic Irony)
The viewer knows about Margot's affair with Richie's best friend Eli long before any characters in the film know. Royal gets wind of it soon after moving back in, but Richie and Margot's husband Raleigh both remain in the dark for a while before a private investigator clues them in. This tension between the viewer's knowledge of the affair and the ignorance of Margot's most affectionate admirers creates dramatic irony.
