Private Ryan
At the start of the film, Captain Miller and his squad find who they think is the right Private Ryan. However, they ironically discover that the Private Ryan they meet is not the right Private Ryan. This unexpected irony creates a greater sense of frustration for the already difficult mission Miller has to undergo.
Captain Miller's last words
Captain Miller, who had been relatively reasonable throughout the film, tells Private Ryan to "earn this" (his sacrifice and the sacrifices of his men). Those words, which hold enormous weight over the young Private Ryan, are ironic because they are directly at odds with Miller's past characterization and his even-keeled temperament.
The film's opening scene
The film's opening scene, which features a peaceful, modern-day cemetery, is ironically starkly contrasted with the film's next scene, which takes place at the D-Day landings. Those landings were exceptionally violent and resulted in profound carnage.
Private Ryan
At the end of the film, an elderly Private Ryan asks his wife if he lived up to Captain Miller's last request and if he was a "good man." Nevertheless, Ryan could never live up to Captain Miller's words without giving his life several times over in battle, reflecting the ironic and oftentimes complex ways war survivors process their grief.