When Finch and Root encounter Larry, Root runs to him regarding his daughter after he denies knowing about 'Northern Lights'. Finch loses his limp as he runs and follows Root.
When Shaw and Reese force the female driver out of her car and before they crash off the overpass, the car is a W221 Mercedes S-Class. After the crash it is a W220 S-Class, an older model.
Finch and Root hijack a private jet to fly to where the nuclear facility is located and Reese and Shaw follow them by flying to Portland. Somehow they arrive at the nuclear facility at the same time which would have been impossible since Reese and Shaw would have to go to airport, buy plane tickets, wait for their flight and then drive from Portland to nuclear facility which is about 4 hours away by driving.
When Reese is given the social security number for the man being held in the shipping container by the Russians, it is listed as 000 - 07 - 9185. This is an impossible SSN, because the Social Security Administration does not issue numbers beginning with 000.
When Reese and Shaw are trying to locate Harold in Central Park, they received another social security number. This one, 000 - 22 - 7814. The Social Security Administration does not issue social security numbers beginning with 000. So this is another impossible social security number.
When the cop brings Carter some coffee while she is in the interrogation room, he puts it on the table and you can hear that it is empty. Also, when Carter puts it down after taking a sip, you can again hear that it is empty.
(at around 30 mins) The date is placed at 2010 showing in slow motion a pan of the street. There is a Fiat 500 in America but the first import year of this car was the 2012 model over 1 year later.
When Reese and Shaw save the man from the shipping container, they leave the laptop intact on their way out despite knowing the police will be there shortly. As the men who were holding the man hostage were recording when they got there, this means Reese and Shaw left evidence for the police that they were there, potentially compromising their anonymity, which they're usually more careful to preserve in such situations.
Reese and Shaw wouldn't have been able to bring their weapons with them since they would be flying across the country.