Thurston's death (dramatic irony)
The subtitle of "The Call of Cthulhu"—"Found Among the Papers of the Late Francis Wayland Thurston of Boston"—clues the reader in to the fact that Thurston has already died by the time the story reaches his or her eyes. The fact that the reader knows Thurston's fate is an example of dramatic irony, given that Thurston frets at multiple points in the story whether he will be killed under mysterious circumstances like Gustaf Johansen and his grand-uncle George Gamell Angell.
"Cthulhu fhtagn" (situational irony)
When Wilcox relates the content of his dreams to Angell, the reader likely expects that Wilcox will reveal some key phrase or image that will help Angell untangle the mysteries of the Cthulhu Mythos. Instead, Wilcox only relates an "unpronounceable jumble" of letters. This revelation is an example of situational irony, in that the gibberish letters only compound the mystery under consideration, rather than clarifying it in some way.
Galvez's testimony (dramatic irony)
In his testimony to the American Archaeological Association, Inspector Legrasse reveals that a fellow police officer named Joseph D. Galvez reported hearing "the faint beating of great wings," and catching a "glimpse of shining eyes," in the swamplands where the men happen upon a voodoo ritual. Legrasse's dismissal of Galvez's testimony is an example of dramatic irony, because by this point the reader knows that the ghastly cosmological forces at hand are likely responsible for the phenomena.
Johansen's death (situational irony)
After Thurston travels to Australia and finally pieces together the fact that extraterrestrial beings are behind the Cthulhu cult phenomena, he travels to Oslo, Norway, to interview Gustaf Johansen, who has died. This is an example of situational irony because the reader expects that Thurston will finally be able to obtain firsthand testimony about Cthulhu from Johansen. Instead, Thurston must learn what he can from Johansen's posthumous diary.
Thurston's manuscript (dramatic irony)
At the end of the story, Thurston pleads with his executors not to let his manuscript be read by anybody. This is an example of dramatic irony because the reader has presumably already read the entire tale by this point, and knows that Thurston's request has not been honored. It also suggests that the reader, armed with Thurston's knowledge, may now harbor the same curse that afflicted him, Angell, and Johansen.