The ending, naturally, especially since it's dedicated to the memory of Jim Henson and Muppeteer Richard Hunt.
A more out-there example, but Steve Whitmire said that he was very nervous about taking over as the voice of Kermit. The night before he began production, he had a dream that he met Jim Henson in a hotel lobby. He told Jim his fears and Jim assured him that he'd do just fine.
Even though they are just playing characters for the movie, it is sort of sweet to see Kermit and Piggy Happily Married, with children.
Clara's reaction to Scrooge giving her and Fred the Christmas presents. She's just so happy that he's there and her surprised smile just sets the whole thing up brilliantly. Doubles as a CMOF for Fred's face.
After Scrooge's conversion, he heads out "to wish Merry Christmas to all the world" and runs into Dr. Honeydew and Beaker (the charity collectors). He asks them to put him down for a very large amount, saying that "a great many back-payments are included." In gratitude, Beaker slips his scarf off and gives it to Scrooge, who is driven almost to tears and laughter by his first Christmas present in no doubt a very long time. Up to this point his attitude toward his own redemption has focused entirely (and quite naturally) on giving, with no thought to receiving. It's like this was the first moment he truly caught on to how love is supposed to be shared.
Scrooge: A gift...? A gift for me?
It's also important to note that until then, all Scrooge wore were muted colors and grey. The red scarf was a Splash of Color for Scrooge.
Gonzo narrates the ending and Rizzo asks him, "but what about... Tiny Tim?", obviously worried. He is elated when Gonzo tells him Tiny Tim did not die! It's a very small but uplifting moment.
Another thing that makes the moment sweet is the fact that when Gonzo says this, he looks directly at the camera and says it very clearly. It's like he wanted to make absolutely sure that any children in the audience knew Tiny Tim would be okay. Awww...
"Bless Us All". The entire song. It's one of the best representations of the Crachit family ever done.
Scrooge remembering his childhood teacher (Sam the Eagle) and old employer (Fozzie Bear) during the climax and stopping by to give them gifts. Fozziwig's happy reaction at Scrooge's arrival is adorable.
"The Love We Found". It's just so adorable and heart-warming.
The entire "It Feels Like Christmas" song can give anyone the warm fuzzies. Even if you're not a Christmas person, you can't help but feel like singing along.
There's a moment during that song which is sweet and a bit funny, where the Ghost does a little dance and Scrooge imitates him.
Also Scrooge peeking in on Christmas Present and the mice celebrating Christmas. He can't help but smile at their sincerity as they sing.