Plot hole: The energy weapons in this story fire a series of small red bullets, which cause numerous problems. For a start, the guns appear to have seven chambers, yet the bullets start from the same point; as they are separate bullets, surely the most minor of movements would mean they wouldn't emerge in a straight line. Later, when the Doctor steals a cart, the bullets are coming quite a distance towards the screen but the bullets don't get any bigger. Also, in the same scene, the Doctor passes through the area they seem to be travelling, yet the bullets just continue to pass in front of him.
Plot hole: The Bandrils not only know of the Time Lords but are capable of connecting to them, yet the Bandril are barely capable of space travel and can't produce their own food. So, the Bandrils are not very technologically advanced it seems, so why do they have ties with the High Council of the Time Lords? And how does either party benefit from such an association?
Plot hole: Peri goes to a fair bit of trouble to bolt the door on the bath house when rescuing the Doctor. When the Rani and the Master arrive, they simply open the door. Now this, in itself, isn't a mistake, as the Master has a special gadget for opening locked/bolted doors, but both the Master and the Rani seem surprised that someone is in the bath house.
Plot hole: The junkyard at 76 Totters Lane seen in "Attack of the Cybermen" was also in "An Unearthly Child", the very first episode of the show. If you consider London's high land value, it is highly unlikely that an ownerless junkyard would still survive twenty-two years... the local authority would have long since issued a "compulsory purchase order".
Continuity mistake: When the Doctor enters the desert room in addition to his coat, he is wearing a brown waistcoat with a white shirt underneath. When the shot cuts away and the back to the Doctor, we see him take off his coat but he is no longer wearing the waistcoat. Later when he collects his coat after the acid bath scene and he starts to look through the wardrobe for a disguise he is mysteriously wearing the waistcoat again.