Running north up Flower Street after the taxi crash, Max grabs a cell phone from a man
standing outside a hotel. When the man objects, Max pulls a gun on him.

That 'robbery' scene was shot in front of The Standard hotel, a popular spot with celebrities.
The hotel is located at 550 S. Flower, in downtown L.A. - on the east side of Flower, between 5th & 6th Streets.
(It's the same hotel they probably used for the Fountain Ave apartment scene.)

Max, trying to warn Anna that Vincent is coming to kill her, runs into a parking garage,
where he phones her from the roof. This scene was shot in the garage for the 777 tower,
(note the "777" sign on the wall) located at 777 S. Figueroa Street.

(The director says that the Secret Service parks their cars on the roof of this garage, and
won't let anyone near the roof - making it difficult for him to get permission to shoot there.)

Vincent goes to the U.S. Attorney's Office to kill Anna, and stalks her through the offices.
Now this should have been at the downtown Courthouse (312 Spring Street). However, the
public hallway scenes were actually shot at the Gas Co. Building at 555 W. 5th Street,
in downtown L.A. (according to Chris W., who works in the building.) 

Since you can see the Staples Center and the Harbor Freeway out the window,
during the scenes inside Anna's office, it looks like those interior scenes were filmed on a set
at the former Unocal tower, which is now part of the new Los Angeles Center Studios,
at 400 S. Bixel Street, in downtown, just west of the 110 Freeway.