Goofs

When The Master is performing the stereotypical "Russian dance," from behind his arms are above his head, however, when the angle changes to a forward view, his arms are in front of him.

Tegan enters the shaft in the basement of UNIT and climbs down a ladder but Cybermen leader Ashad hears her when he is on the upper floors and Cybermen break the wall to try to grab her. Tegan should be far below where Ashad is located.

UPDATE: Tegan doesn't enter the shaft in the basement. Kate Stewart and Tegan started on top of the building and they need to get to the basement.

When the UNIT guards apprehend The Master, they handcuff him behind his back. In the next scene, his hands are cuffed in front of him.

When The Master leaves the TARDIS, he is wearing the 13th Doctor's clothes. A few moments later, he is wearing a mashup of the 5th, 6th, and 7th Doctor's clothes and remains in that outfit for the remainder.

UPDATE - The Master tells Yaz he needs clothes and goes off to change, telling her if she touches the TARDIS he will kill her. He is then in the new costume for the remainder.

As the Daleks initiate the volcanic eruptions, The Master/Doctor is wearing the earring/cuff on his left ear. Seconds later, during the same scene, the cuff is gone.

Among the famous paintings said to have been taken out of public view due to the Master's defacing is The Last Supper by Leonardo da Vinci. the Last Supper is painted on a refectory wall and cannot be moved.

Being as close to magma flows and volcanic gas as several characters are should have killed them.

The traitor Dalek has a visible bit of damage on the upper right side of its eye stalk, however, in one closeup (after The Doctor has arrived to speak with it in the magma caves), its eye stalk is completely intact. A closeup of a random Dalek seems to have been substituted instead of the damaged traitor Dalek.

When the Doctor is speaking with the Dalek holo inside the TARDIS, the "Police Box" door label is not reversed.