When Frank and Lilly are disrobing on their way to the bed, you see all of their gear fall to the ground piece by piece and then their clothing. Lilly turns out the lamp, wearing pants, then lies down wearing a slip.
When the agent is checking the pack of photos against guests arriving for the dinner, Mitch's photo is a single frame. He then looks at Mitch, doesn't recognize him, and shuffles the now two-frame photo to the bottom of the pack.
Frank and Lilly eat ice cream on the steps of the Lincoln Memorial in the late afternoon. The east-facing steps are shown to be completely in shadow in long shots from the front and rear, but in close-ups their faces appear to be sunlit.
After Lily steps away from Frank as he's playing piano, she walks across the room, exits the bar, then crosses the lobby to elevators. All the while Frank's piano playing continues. Once it stops, he clears the same distance in less then three seconds.
Lilly's gown during the party scene would be inappropriate for a female Secret Service agent, as it would prevent her from performing her duties should there be an attempt on the President's life. In those situations female agents instead wear dress pants and more practical shoes. (With the gown, there is also the problem of where to hide the service weapon.)
In the scene where the Secret Service counter-snipers attempt to get a site on Leary, it is reported that it is too dark for them to see inside the elevator. Secret Service counter-snipers are equipped with night-vision goggles to allow them to see targets through darkness.
When we see Leary's view through his binoculars, his view forms the figure eight, but if binoculars are correctly adjusted, they form a circular view. This is a common error in movies, and is often made deliberately in order to show the viewers that now we see the scene through binoculars.
As Mitch Leary assembles his composite-based gun at the Presidential Dinner, it is obvious that the thumbscrew is a painted metallic piece that would have been caught in the metal detector as he entered the event.