According to Michael Anderson, the old man's buttons are United States pennies. He made makeshift buttons out of them because he couldn't find any real buttons.
On the day of shooting, director Michael Anderson and producer Saul David decided that Logan should look more "casual" for the first scene in his apartment. Costume designer Bill Thomas threw together Logan's black house robe in about two hours while the set was being lit. Michael York kept the robe as a souvenir after filming.
After Logan and Jessica enter Washington, they are seen wading in the reflecting pool on the National Mall, approaching the Washington Monument from the west, with the Capitol visible behind. That would put the Lincoln Memorial behind them. Yet the next scene has them climbing the steps of the Lincoln Memorial they would have just passed.
When Logan and Jessica first emerge from the ice cave and see the sun, they have no idea what the sun is. Later, when they attempt to re-enter the city at the water falls, Jessica asks what they're for. Logan says that he thinks it's part of the city's power generation system. He says "The ocean tides are turned into energy somehow". If they no longer have any knowledge of what a "sun" is, it's doubtful they know what an "ocean" is, and especially doubtful they understand the concept of "tides".
When Logan first meets Jessica in his apartment, the level of his drink changes from full to half full and back after he takes a drink, but does not add any liquid to the glass.