When young Ebenezer and Isabelle are on the horse-drawn carriage in the countryside (shortly after the Fezziwig ball), there is a white car crawling its way along a country lane in the far background, midway between the centre and right-hand side of the screen.
During the singing of "Christmas Children" a lady shopper is seen loading up a basket filled with grapes and a pineapple. Considering the time that the story takes place (circa 1843), fresh fruit like this would have been impossible to acquire in December.
During the reprise of "Thank You very Much" at the end of the movie, the crowd sings and dances their way past the booth of the Punch and Judy man and the camera stops to focus on him. If you look just to the left of his booth, you can see the actors stopping and turning back since they are now "off-camera." But, the next shot after the Punch and Judy man shows them continuing down the street.
After Scrooge wakes up to discover it is Christmas day, he slides down the banister in his home. The banister is very dusty and when he first jumps off, one can see a glimpse of the four inch wide line of dust/dirt down his night shirt. But the very next scene, he walks into the street in his nightshirt, and there is no line of dust/dirt down his front.
When we first meet Tiny Tim, he and his sister are looking into a toy store window. If you look closely at the bottom right corner of the screen as the camera zooms in on Tim, you can see the shadow of the camera on a dolly.
While with the Ghost of Christmas Present, Scrooge peeks into a home, wiping the frost and condensation off the window. Setting aside the question of whether an incorporeal spirit could do this, frost and condensation collect on the warm side of a window, not on the cold side, where Scrooge is.
In all the night street scenes, the shop windows are too brightly illuminated to be made with just candlelight or gas lanterns, which would have been the only sources at the time this story takes place. They are obviously electric lights, something only available far in the future.