Anvilicious: Shepard's press conference speech. Let's see: the ACLU is supremely good and noble and anyone who doesn't belong to it doesn't believe in the Constitution; "going after the guns" is GREAT for fighting crime; flag-burning is a symbol of freedom; and there is absolutely no reason anyone should have a legitimate problem with Shepard's relationship with an environmental lobbyist even though in the exact same speech he's just talked about going through with the legislation she's been pushing. Uh-huh.
To be fair, the issue was being obscured by Moral Guardians crusading against a widower having a sexual relationship while in office. Any objections to the President's policies or issues regarding Conflicts of Interest were drowned out in the dog and pony show being generated by the opposition which is, sadly, Truth in Television.
Hilarious in Hindsight: In January 2012, then-Australian cabinet minister Anthony Albanese plagiarised lines from this film in a speech slamming the then-Opposition Leader: "In Australia we have serious challenges to solve and we need serious people to solve them. Unfortunately, Tony Abbott is not the least bit interested in fixing anything. He is only interested in two things: making Australians afraid of it and telling them who's to blame for it." This will undoubtedly affect the way many Australians see the film in future, as will the line, "Well I sure hope "the Sydney issue" refers in some way to a problem we're having with Australia, because if it's anything else-".
Slightly less hilarious now the Liberal Party under Tony Abbott has regained power, arguably as a result of doing exactly that.
Also, a lot of lines become this for fans of The West Wing, in particular "Someday someone's going to have to explain to me the virtue of a proportional response."
Strawman Has a Point: While it was not stated in the movie as clearly, with the focus on the relationship and the obvious comparisons to President Clinton's troubles, what President Shepard was doing was somewhat unethical as seen from the outside. He was in a relationship with a lobbyist that he was also working with at the time on a major piece of legislation. If a real politician did this he would justifiably be attacked in the press for this.
In the Fridge Brilliance page, it's mentioned that Rumson never attacking the conflict of interest issue is actually good for his campaign. A legitimate issue like that would resolve the scandal and play into the fact he doesn't have any real charisma (from what we see) or good arguments to oppose the President's policies (even if they are worthy of being challenged).