After Mamie ties Emmett's necktie, it is not tucked under his collar in the back. Later in the same scene, it is perfectly tucked.
The TV screens, although they have rounded corners and display low-quality black-and-white video, are obviously cleverly disguised modern LED screens: the picture is stable, there is no noise, no snow, no wavering sound.
The 15 September 1955 cover of "Jet" magazine featured Beverly Weathersby on the cover, not Emmett Till and his mother with her fiance' as depicted in the film. The article about Till's horrendous murder, including photos, was featured in the interior of the magazine.
In the barbershop scene, one barber is using an Afro pick, something that wasn't invented until the 1960s.
When Mamie walks past Polk's Barber Shop at 17:44, the front window advertises "unisex and nails". The term "unisex" wasn't coined until the 1960s.
The flashlights that the sheriff and his deputy use are LED--decades before their invention.
In the beginning, after Emmett sings the Bosco chocolate-syrup jingle, he gives Gene a high-five, which was not part of African-American or American culture in the 1950s.