When Charlie Brown is learning to dance using the foot markers laid down by Snoopy, he counts; "1, 2, 3, 4" but as he says "4" he puts his left foot down on the marker that says "3". Of course this is what we'd expect from Charlie Brown ...
While Snoopy traverses the Christmas lights in Peppermint Patty's window, she says "Yeah Chuck, your crazy dog is over here again!" over the phone, contradicting the running gag in the comic strip, TV specials, and movies (including this one) that she believes Snoopy is a "funny looking kid with a big nose" rather than a dog.
Charlie's hands get covered in ink while writing his book report. He then wipes his hands on his shirt creating the characteristic zig zag pattern. However, in all subsequent shots the pattern disappears and his shirt is again a solid color with no pattern.
When Charlie Brown goes to bed after writing his book report, while he is getting into bed we can see some notes beside his window sill. However, when Sally comes in and wakes him a few seconds later, the notes have vanished.
Charlie Brown is forcefully towed by his kite across the frozen lake, even though there is zero wind (as shown by the gently falling snow flakes). However this is essentially in keeping with the style of how snow was drawn in the original animation and comics, and may be a deliberate choice by the filmmakers.
The Red Baron's triplane lacks the Iron Cross symbols that the real plane had in both Linus's original model and in Snoopy's fantasy sequences. Linus may have forgotten to include the Iron Cross symbols or had not yet painted them on, while the Red Baron's plane in Snoopy's fantasies was based on the model plane because it hit him while he was trying to climb into the classroom through the window.
Linus is a year younger than Lucy and Charlie Brown, and would not have been in the same class as them. This is because he skipped a grade, which was not an uncommon practice when the comics were written.