Ask HN: Is there a free list of cities mapped to their tz database time zones?

We (https://opencagedata.com/demo) return that as part of our geocoding results. If you have only a couple of hundred cities it's still within the free trial plan

        "timezone": {
          "name": "Asia/Amman",
          "now_in_dst": 0,
          "offset_sec": 7200,
          "offset_string": 200,
          "short_name": "EET"
        },

As detaro wrote the timezone-boundary-builder project is the best data source for the shapefiles of timezones.

For a list of places the files on http://download.geonames.org/export/dump/ have a column for the timezone name. Creative Commons Attribution 4.0 License.

There are places, usually unhabitated who don't have a good timezone name, but just 'GMT-10' or such. With cities you likely don't encounter this. Short names, e.g. EET, aren't unique. There are several who share the same abbreviation.

One part of the puzzle are projects like this, giving you location to timezone: https://github.com/evansiroky/timezone-boundary-builder

Mapping city to location is a classic geocoding task, for which there's APIs and tools around.