When the Ballplayer is at the bar next to a wall calendar, in the medium shots the calendar page is for June 1954, but in a close-up the page says March 1954. The mistake would seem to be the close-up because there is a night-time baseball game on the bar's television and the baseball season didn't begin until April.
When the Senator first sits down in the hotel room, there is a stack of papers on a table to his right. The papers are much more neatly stacked in the longer shots than in the close-up.
The movie takes place in 1954. In one scene, a man is in a bar watching the World Series. The TV announcer for the game starts talking about "Campanella, Hodges, and Berra" being involved in a play. In 1954, those three players played for Brooklyn Dodgers and NY Yankees; however, the 1954 World Series was between the Cleveland Indians and NY Giants, therefore none of those three even played.
At 0:03:23, the scene on the bar TV shows a ballplayer sliding into third base ahead of a throw from left field. The announcer says, "and there at the plate, he steals home!" Clearly the ballplayer is sliding into third base. And since the baseball is coming in from a left field, it's not an attempted steal.
When the Actress buys balloons at a newsstand, several vintage magazines are prominently displayed, but mid-1980s magazines can also be seen, including an issue of PC World.
The flag flying outside The Roosevelt hotel just before the Senator first goes up to visit the Professor is a 50-star flag instead of the 48-star flag that would have been there in 1954.
When "Actress" (Theresa Russell) is in the backseat of the car after leaving the news stand, there are several shots where modern cars can be seen. Most noticeably between approximately 20:45-20:49.