The first time Jo picks up a pen to write, she uses her left hand, but when she is signing her name to the novel "My Beth", she uses her right.
When Jo gives Beth the music box, Beth opens it, then in the next shot, she opens it again.
When Mr. Bhaer is reading Jo's writings, the pinky ring disappears and reappears on his left hand.
When Jo checks in on her sleeping sisters, Amy has a clothespin on her nose, her mouth is closed, and yet she still must be breathing. This isn't a goof though, as a person can "sneak" breaths through their lips while it appears their mouth is closed. Also, the clothespin likely was not sealing her nose completely.
When Laurie stalks off after Jo rejects his marriage proposal, his shadow is visible on the backdrop that is supposed to make it look like they are sitting outdoors.
At breakfast with popovers, they are served coffee. The actress holds the handle with a pot holder, yet holds the spout end with her bare hand. Obviously the pot does not contain hot coffee.
When Beth plays the piano for the melodrama, she wiggles her fingers on the piano keys but doesn't rapidly alternate notes, which would be required to produce the tremolo that is heard.
As Marmee is watching the girls leave to shop for Christmas presents, a modern zipper is visible up the back of her dress.
At the ball, one of the tunes being played is the Pig Song from Johann Strauss's "The Gypsy Baron" (1885).
Marmee is checking on the girls to make sure they are asleep. She picks up the "oil" lamp at the top of the stairs and the electric cord is visible running along her sleeve for a moment.
Laurie asks Jo how many lumps she wants in her tea, and she says three. It is seen to be normal sugar, and Laurie puts four spoonfuls in the tea.