There's a scene where Mildred Loving is shown washing dishes at home, and the dinnerware appears to be made of Corelle. This brand of dinnerware was not introduced until 1970, and the scene in question would have been mid to late Sixties.
When the Lovings are walking with their lawyer to the Caroline County courthouse in Bowling Green in January 1959 to enter their guilty plea, they walk in front of the Bowling Green Post Office. The sign on the Post Office building includes the Bowling Green ZIP Code (22427). ZIP Codes were not introduced until 1963.
The female reporter who is interviewing Mildred in their house is recording her audio on a Nagra 4.2 tape recorder. Tha Nagra 4.2 was introduced in 1971, while the scene takes place in the mid-sixties.
At the beginning of the movie, when the two cars are racing, one of the drivers is stepping on a brake pedal with a disc brake emblem. Disc brakes were not mass produced until the mid 60s.
Early in the film a burger joint is show in the background. Its signage is in the Apple Chicago typeface, which was not created until the early 1980s.
When Mildred finishes sewing part of a garment, she yanks it away from the machine, then she reaches for the scissors. When sewing on a machine, one never yanks at the fabric, at least not without carefully pinching the area just sewn. Otherwise, the area just sewn puckers, plus a lot of thread is pulled off the spool and the bobbin, and thus wasted.
The film made a good effort to recreate the scene of one of the iconic photographs from Life magazine, where Richard is lying on the couch with his head on Mildred's lap. However, in the original Life photo (shown during the credits), Mildred is holding a cigarette. Mildred was not shown holding a cigarette in the recreated scene.
While appearing in front of the Virginia judge the second time for violating parole, their lawyer asks for leniency. When he walks in his tie is askew and top button on his shirt is undone. After the judge agrees to grant leniency, his shirt is all buttoned up and tied nicely.
Regarding the drag race scene, the brake pedal shot indicates that the car had a 2-speed Powerglide transmission as there was no clutch pedal. Also the shift lever was in the Park position making it impossible for the car to move. This would be the only Chevy small block ever hopped up with a "slip and slide Powerglide" transmission. They all had stick shifts, later with close ratios.
During the turkey dinner at Mildred's parents' house, both Mildred and Richard use their utensils in the European/Australian style. By contrast, everyone else at the table use their utensils in the American style.