Woodward and Bernstein as Rivals (dramatic irony)
Woodward and Bernstein do not begin the film as the prolific duo they would grow to be. There is instant resentment and even rivalry between the two at the start of the film, even featuring the relentless Carl Bernstein stealing and rewriting Bob Woodward's story. The dramatic irony for the audience lies with the fact that we know they will soon become friends as the iconic duo "Woodstein," but that we don't not how it will begin.
The unnecessary Watergate break-in (situational irony)
Nixon was already so far ahead in the polls and McGovern was so famously self-destructing, that the bugging of the Watergate office would likely not have lead to any different outcome than Nixon's inevitable landslide victory. The paranoia and unscrupulousness of the administration leads to overcompensating and breaking laws. The situational irony of Nixon's unconstitutional surveillance schemes from 1971 only comes to light in 1972 when he is vastly leading in the polls.
Nixon's standing at the beginning of the film (dramatic irony)
When the film opens, Nixon seems invincible. He enters to address a joint session of Congress to a wild, standing ovation. Yet, by the end of the film, he is humiliated as the only President in U.S. history to ever resign the office of President. Power is truly fleeting; one moment you're up, the next you're down. This scene—and indeed, the entire film—plays out as a dramatic irony, since the audience knows that this seemingly-invincible man will soon go from being exalted to humiliated.
"Crazy Cubans" (situational irony)
Rosenfeld says in all likelihood, the Watergate break-in was just a result of "crazy Cubans." The Post's initial underplaying of this earth-shattering scandal highlights just how unprecedented the scandal was. This situational irony highlights the contrast between what Rosenfeld's instinct was in the moment and how in fact the "crazy Cubans" were just the tip of the iceberg.
