Blooper / Off Model: Several animation errors exist, one of the more obvious being the fact that the words on Lucy's psychiatric help desk change twice in less than a minute. Among others:
Lucy accidentally calls fear of cats "ailurophasia" instead of "ailurophobia."
Sally trips over the line "All I want is what I have coming to me. All I want is my fair share." This is due to the fact that the girl who voiced her was so young that she hadn't learned how to read yet and had to be fed her lines word by word, and in some cases, syllable by syllable.
Channel Hop: After airing on CBS for 36 straight years, the special moved to its present network home of ABC in 2001.
Christmas Rushed: Schulz was given six months to complete the special in time for the holiday season, and animating didn't start until the third month in; the results are quite conspicuous.
Edited for Syndication: In its original 1965 broadcast, this cartoon clocked in at just over 26 minutes. Nowadays, American TV networks have a lot more time devoted to commercials than back then. So for most of its subsequent airings, several scenes have been cut short (the first dancing scene) or cut out entirely (much of the beginning). Even in its first VHS release, the scene of the kids throwing snowballs at the can on the fence was cut out. It was restored for its second VHS release and was re-instated in network broadcasts in 1997 (with the Coke can redesigned.)
Coca-Cola was the special's original sponsor, and a brief scene mentioning this was animated for the broadcast. Once their sponsorship ended, that scene had to be excised and was thought lost for decades until a film reel containing it was discovered, digitized, and uploaded onto the internet (rights issues prevent it from being on the DVD cut, though every other scene cut over the decades is restored).
Interestingly enough, Coca-Cola offered to restore the cut sponsor tags when the special was remastered, but Schulz Studios turned them down, believing A Charlie Brown Christmas to be perfectly fine without them. To be fair, they would undermine the anti-commercialism message of the story.
The recent showings on ABC (which generally broadcasts both a cut version and the uncut version over a given holiday season) have taken further cuts, removing such iconic moments as Lucy pestering Schroeder over the proper playing of "Jingle Bells," to even cutting poor Shermy's only line.
Executive Meddling: CBS executives had some issues with the special as it was originally put together. They objected to:
The Scripture quotation that Linus recites to explain the meaning of Christmas.
Using actual children to voice the Peanuts characters.
In other words, nearly everything that makes this program a timeless classic. Fortunately, Charles M. Schulz and Bill Melendez stuck by their guns and were vindicated the moment it aired.
Magnum Opus Dissonance: Nobody behind the project thought it was any good when they finished; Lee Mendelson and the rest of the team felt they "ruined Charlie Brown" when they looked at the final cut before it aired.
It apparently took a while for Charles M. Schulz himself to realize how iconic this special had become. In the late 1980's he fretted that he hadn't produced his "own Citizen Kane" and invested a good deal of time and money on his intended masterpiece, the flop live-action/animated combo It's the Girl in the Red Truck, Charlie Brown. Any fan could've told him he'd already made his Citizen Kane way back in 1965. (For added irony, Orson Welles didn't consider the actual Citizen Kane to be his masterpiece.)
Network to the Rescue: CBS made some cuts to the special in the '90s because shows made more room for commercials by then (see Edited for Syndication above). When ABC acquired the rights in 2001, they blocked out a full hour for the special so that it could run uncut, commissioning Charlie Brown's Christmas Tales to fill the extra time. (ABC does air cut half-hour versions some years, but the full version will be shown at some point during the season.)
The Red Stapler: The trope was reversed for this special considering that it destroyed the market for Aluminium Christmas Trees so completely that later generations have to be reminded that such Christmas decorations actually existed.
Replicas of the little flimsy tree Charle Brown gets are now being sold.
What Could Have Been: Apparently when the project was started, CBS just wanted this to be a regular goofy Peanuts short that just happened to take place at Christmas time. They weren't expecting the shots at commercialism and Linus's big speech. It's a good thing the special didn't go this direction, too, because it wouldn't be nearly so memorable otherwise.