Factual error: The team is discussing the location of the Widmark hotel which the detective says is downtown. Beckett then looks at the map and locates it only 5 blocks from E 72nd street, which is UPTOWN in NYC. Furthermore the map shows E 72 intersecting Broadway which is on the west side of Manhattan. This entire mapping exercise is incorrect.
Factual error: When Castle and Beckett go to talk to the victim's assistant, there is a chalk outline on the floor where the body was discovered. Although it's common in police dramas, chalk outlines are not actually police procedure, because it contaminates the crime scene.