Dramatic irony exists when the reader, or the viewer knows more about the story than the characters.
In Chapter 1, Agatha Christie introduces the reader to the characters, each has been invited to the island under unusual circumstances, while each character seems to accept the invitation without question, the reader becomes suspicious of why this diverse group would be gathered together at the same time.
So the reader senses that each of them is being deceived or tricked, therefore, the irony is illustrated through the fact that the reader understands the truth to be different that what each character believes it to be.
This has to do with one of the themes that runs through the book which is appearance versus reality. Things are not always what they appear to be.
"The action begins under a cloud of deception when Judge Wargrave, under the guise of the mysterious Mr. Owen, lures the group to Indian Island. The deception continues after the voice on the recording accuses each of a crime and they all deny any responsibility. Wargrave’s confession reveals the final deception when he exposes his faked murder and his own true nature."
The irony is subtle, and a bit confusing, since the reader really does not know what to make of this group of diverse characters who are all traveling to Indian Island for different reasons. It is through understanding that each person has been personally invited by a long lost friend, acquaintance, or someone who is heading to a job that the reader suspects that something mysterious is going on, and of course, because it is an Agatha Christie book, the reader can expect some twists and turns in the plot.