While Holden and Debbie are watching Dog Day Afternoon (1975), there is no flicker seen on the projected screen, and the movie frame edges are visible within the screen. They are watching the movie from a digital projector. Additionally, a 35mm projected film would fill the entire screen and be masked by curtains on all sides to hide the rough edges of its aperture plate. Digital projection was not used publicly until 1998.
(at around 37 mins) Bill's coffee cup is obviously empty.
Classrooms at the FBI Academy are shown with whiteboards, which didn't gain widespread use until the 1990s, instead of chalkboards.
As Ford and Tench are driving away from Iowa, the song playing on the radio is "Crying". Even though the original version by Roy Orbison was released in 1961, the version playing was from Don McLean, which was on his 1978 album "Chained Lightning" and wasn't released as a single until late 1980.
As Ford and Tench are driving to Iowa in 1977, the song on the radio is "Hold the Line" by TOTO, which didn't come out until October 1978.
(at around 32 mins) A modern "recycling logo" can be seen underneath a brown lunch paper bag sitting on a table.
(at around 20 mins) When Holden (Jonathan Groff) and Debbie (Hannah Gross) are on the dance floor, the music was added in post-production. During filming, the music was played through earpieces, and they can be seen in Groff's ear as he says "easier than this." The one in Gross' ear can be seen as Holden says, "Would you please stop?" moments later.