Q:

Why do address forms ask for the US state?

Inspired by Why do credit card forms ask for Visa, MasterCard, etc.? - 
Why is it standard for web sites in the United States to ask for ZIP code and state? 

Asking for the state is redundant. The ZIP code already provides an unambiguous identification of the address.
Asking for the state is inconvenient. It's usually a drop-down with 50 entries, and is often not caught by the browser's autofill feature (at least not in Chrome).
True, the state is basic required information  when shipping with the USPS. Also, it may be relevant for tax and other regulations.
However, it would be easy to fill in automatically based on the ZIP
code the user entered, e.g. using USPS' API, or plain old Google Maps.

Why does everybody and their dog still force you to enter a state?

A:

ZIP codes are not unambiguously assigned to states. ZIP codes are assigned strictly on the basis of how efficiently the mail can be delivered.  Some codes span multiple states, often when a small border town straddles two states. While a parcel with the correct ZIP code will be sent to the proper address, the customer wants their state to be recognized.
See http://www.unitedstateszipcodes.org/ for more details.

A:

Asking for the state is redundant

Exactly, it's redundant:

redundant (adj.) Duplicating or able to duplicate the function of another component of a system, providing back-up in the event the other component fails.

If you get the zip-code wrong, the letter can still be delivered if the street and state are correct.

