When Donna and The Doctor are first running down the tunnel away from the TARDIS, the door is open but in the next scene it is closed.
The first time we see 'Santa' driving Donna away in the taxi he has gauze for eyes. In the following scene as Donna knocks off the mask, the eyes are solid and painted on.
Everyone throughout is dressed as if it is warm, and the trees in the background on the motorway chase are lush and green. That's not London on Christmas eve.
4,000 miles to the center of the earth means that no light would be visible at the bottom of the borehole, and the water would take 38 minutes to reach the core.
As The Doctor runs down the stairs after Donna stops him, a crew member is seen under the stairs looping coil, probably to control the water flow.
As the wedding begins, the organist begins to play Felix Mendelssohn's "Wedding March" as the bride walks down the aisle. However, this piece is traditionally used as the recessional piece, while Richard Wagner's "Bridal Chorus" ("Here Comes the Bride") is used for the processional.