How can I get the time zone of a US ZIP code?

TimeZoneOffset[Entity["ZIPCode", "94402"]]

-8.

This will do what you want:

TimeZoneOffset[GeoPosition[Entity["ZIPCode", "94402"]["Coordinates"]]]

-8

Each zip code entity has coordinates associated with them, and TimeZoneOffset can take a GeoPosition and return the timezone offset from GMT.

It should be noted that kglr's answer is the shorter (and more graceful) form of this answer - they are doing the same thing under the hood.