The building used in the outside shots clearly was the Roosevelt Hotel, but the Roosevelt (and all outside shots of the building in the movie verify this) has only 18 floors, and the movie premise is a hotel of at least 24 floors.
In the flashback at the beginning (at Sing Sing Correctional Facility, a month earlier), while talking with the psychiatrist you can see that Nick has plaster on his arm, due to the fight we saw before. In the next scene he gets a visit from Mike, but then his arm is no longer plastered.
After Joey opens the panel beside the vault door, Angie looks into it and we see the vent beyond goes on for a fair distance. However, later in the movie, we see into the vault itself and it's evident the layout of the vault and corridor couldn't possibly accommodate a vent of the length shown.
When Joey and Angie reach the corridor with the heat sensor they spray it with a fire extinguisher to chill it. When the Englander security team arrive, they pass through the corridor and into the vault. When they gain access to the vault there is an extended shot of the frozen heat sensor inside the vault, with a drop of melting water forming to potentially drop onto the head of security. Joey and Angie were still hiding at that point and hadn't yet reached the vault themselves, so couldn't have frozen the sensor.
When Joey stops the fan and bends the blade down so that he and Angie can get to the elevator shaft, the bent blade is pointing up at a 45 degree angle. As Joey and Angie are rigging their ropes to drop down the shaft, the bent blade changes from 45 degrees up to 45 degrees down, and back a few times.
After Lydia gets handed the note about the room having been swept for fingerprints, there's a shot of Nick standing beyond the narrow part of the ledge. After a brief shot of her, in his next shot he is suddenly standing back on the narrow part itself.
When Lydia Mercer is talking to Ackerman in the hotel room, she's holding an orange coffeecup. In some shots you see the front side of that cup with a logo. In other shots you see the back with a barcode.
On the rooftop of Englander's building both Joey and Angie are seen reaching the bottom of a ladder before Joey speaks to Nick from his phone. In the next shot the camera is positioned up with Nick and we see Joey at the bottom of the ladder but Angie only half way down.
A short while after Lydia first offers him a cigarette, Nick moves almost to the corner of the building to communicate with Joey and Angie. In a cut to ground level with Suzie Morales and her cameraman, there's a shot up to the ledge where a figure clearly stands right next to the open window. In the next close-up, Nick's back near the corner.
When Joey finishes cutting into the wall with an electric concrete saw during the diamond heist, the blade is spinning in the opposite direction (away from him); Blades actually work by spinning toward the operator, not away.