When the Doctor and Clara first meet Robin Hood, Clara exits from the Tardis and closes the door behind her. A few seconds later the Doctor is on the log bridge and the door of the Tardis is open. For the rest of the scene the door is closed again.
During their fight, Robin Hood cuts off The Doctor's topmost coat button. It is gone for the Doctor's trick on Robin, but when The Doctor and Clara are looking into the stream for Robin, the coat button has miraculously sewn itself back on.
At around the 3:40 mark, when they first meet Robin Hood, Clara exits the Tardis and closes the door. In the next scene where the Doctor and Robin are facing off, the door of the Tardis is ajar.
The dungeon room in which the Doctor, Clara and Robin are locked is far larger than one would expect for a room of that function. It only contains four posts for securing prisoners. A prison room this big would contain more spots to put people or it would be smaller. It would also be in likely to contain exactly the number needed for the number of people the Sheriff was hoping to catch.
The trap that was the archery contest was hoped to capture all of Robin's men, not just Robin himself.
When Robin, the Doctor and Clara are chained up in the dungeon the post to which the Doctor is chained moves in response to the Doctor moving.
When the robots are shown in orbit the Earth that is show is present day earth, not 1190-era earth. Clearly visible is the Dutch Flevopolder which was reclaimed around 1950.
Red and Delicious apple was not developed, According to Wikipedia, till the 1880's. It did appear to be the apple he took and ate after he decried the bantering at Robin Hood's camp.
Alan-a-Dale refers to the Sheriff of Nottingham as a "jackal". The word was not attested in English until the 17th century, and jackals were unknown in Britain in 1190.
When the sheriff shoots his first arrow he doesn't aim properly.