There is a point where a Dalek is looking at Mr. Diagoras and his eyestalk is moving up and down, but when it cuts to the view from the Dalek, it is still.
While on his way to meet the Dalek leader, Mr. Diagoras puts his right glove on twice.
When Mr. Diagoras is first being merged he has gloves on, but when he emerges as a hybrid, the gloves are gone.
The Daleks are to top off the Empire State building with a metal called Dalekanium, but in the John Pertwee story 'Day of the Daleks' from 1972, Dalekanium is an unstable explosive.
Set in November 1930, yet the Empire State Building's spire is towering over the Chrysler Building. In reality, the ESB did not get taller than the Chrysler until April 1931. But, with the Daleks pushing to get its construction completed, it would have been finished sooner in the Doctor Whoniverse.
When the people in Hooverville are meeting around the campfire, it's raining. However, everyone's clothes and hats are completely dry.
The Statue of Liberty is shown with the all gold-leaf surface torch that was put in place in 1984. In 1930, the torch was made of glass panes mounted on a copper frame.
The light that the Doctor points at the alien organic matter when he comes to analyse it is a Strand Pattern 765. These were not manufactured until 1972.
The Hooverville in the middle of Central Park was first established in 1931, not 1930.
According to their case insignia, Mr. Diagoras gets into the elevator with Dalek Thay but gets out with Dalek Jast, yet according to closed captioning, it was Dalek Caan.
The Empire State Building can be seen from Liberty Island (where the Statue is). But the view seen by the Doctor and Martha looks like they are across from Midtown Manhattan rather than the Financial District, as it should be.
All over the backstage area of the supposed "Broadway" theatre, highly distinct conical fire extinguishers can be seen. These were made by an English company named Minimax. While they were being exported to a number of countries during the 1910s and 20s, they were not in use in the USA at the time.
The fake newspaper seen on various occasions has "city" written as "citiy" in the strap under the headline.
While trying to escape from the Daleks in the Empire State Building, one of the Americans refers to the elevator as a "lift," the British synonym.
While in the sewers talking to Martha, Frank claims to be from Tennessee, but his accent is wrong for that region. It is closer to being a Texas accent, but it is obviously an affectation.