Does "The Doctor, The Widow, and The Wardrobe" tie into the overall storyline, or can it be skipped?

Yes, you can freely skip all the main storyline of this episode. It's a classic one-off Christmas special, involving characters that never appeared in any other episode. There's no recurring villains like Daleks or Cybermen, and no worldbuilding or scene-setting that's used later.

The only scene which does fit into a (sort of, to the extent there is one) overall series arc is the very last scene, after the main adventure with the one-off characters is over, when

Even that's not really important - just a bit of emotion, no big reveals or character development - but it's only 2 minutes long and you can easily watch it on Youtube: