Alternative Character Interpretation: Was the Postman crying at the end because he was defeated or were they Tears of Joy? If you look below, one of his reasons for hating Christmas was being put on the naughty list, so it's possible that he was touched that the penguins gave him a gift, even if it was raw fish.
Did Fido actually mishear Tim's words, or did he actually hear it just fine, but deliberately chose to lie? While it's true that it would be hard for either Olive or Fido to hear through the walls of the doghouse, it doesn't change the fact that Fido wasn't so far from it that he couldn't have at least made out what Tim was actually saying. However he knows that while Tim was certainly apologetic and that the issues between him and Olive could be resolved, he also knows that this won't change the fact that Santa will not be able to make his Christmas run without another reindeer...such as his good buddy Olive perhaps.
As such, he decides to pull a rather cruel (albeit well intended) gambit by intentionally lying to Olive about Tim's words, and thus achieves two things: He ensures that Tim will have to truly understand how hurtful his earlier behavior was and thus truly appreciate Olive more, and Olive will be fully dedicated to try to get to the North Pole to help out as a replacement reindeer. The latter point also comes with the benefit of Olive coming to appreciate her time with Tim and Fido once her mission was complete. Granted this plan, if true, dips into Fridge Horror territory when you take into consideration the numerous ways it could have failed (Tim wasting his Christmas trying to find Olive, Olive potentially dying on her way to the North Pole and thus never making it, or even if she did make it deciding not to return to Tim out of pain from the "misheard comments"), thus highlighting how shortsighted such a plan was (although then again Fido IS a flea; long term thinking doesn't seem to be his strong suit). Fortunately due to this being a Christmas special, it all works out in the end.
Fridge Logic: The Mondegreen which starts Olive on her journey was changed in the adaptation likely for this reason: It's a major Contrived Coincidence everyone Olive heard singing happened to be on that particular line and seemingly stopped at that point, or she would have realized it wasn't about her.
Heartwarming in Hindsight: Drew Barrymore's character (the lead heroine, no less) and first-born daughter share the same first name, making this special even more adorable.
Jerkass Woobie: The Postman. The reason he hates the Christmas season is because it greatly adds to his workload. People who have to consistently work during Christmas can probably sympathize with him. Although, it is implied that the true reason is because he's mad at Santa for putting him on the naughty list.
Memetic Mutation: "Bah, Bug and Hum" is modestly popular enough to spawn a few Fan Vids featuring villains from other properties.
Moe: Olive manages to be one and a strong character at the same time.
Strawman Has a Point: While the Postman is already a Jerkass whp went too far by trying to ruin Christmas for everyone, his dislike for the holiday still holds water. Since Christmas seasons greatly adds to his workload, but we the audience are supposed to root for Olive.
Tear Jerker: The scene where Olive thinks that Tim is getting a new dog and replacing her (unbeknownst to her, he was apologizing for snapping at her).
The Woobie: Olive wants to save Christmas despite people, the weather, and bad luck impeding her quest. Viewers can't help but feel sorry for her when she almost feels like giving up.