Assuming that the table is called t and the column is called x:
SELECT *
FROM t
WHERE exists(
SELECT 1
FROM json_array_elements(x#>'{general,tags}')
WHERE array_to_json(array[value])->>0='first_tag'
);
This does not use jsonb or other newer stuff so it should work on 9.3. See also sqlfiddle.
The idea is to use the json_array_elements function, which converts a json array into a sql table. That table has a single column called value of type json.
In this case, we use json_array_elements in a subquery to iterate through the list of all tags. One complication is that value (which here represents a tag) is of type json, so we have to convert it to a text. Unfortunately, postgresql doesn't have a function to do that directly, so we have to convert it into a single element array and then extract that single element as text. Then we can compare that text with the tag we are look for (first_tag in the above example). The result of this column is not significant, we are only interested in whether it is empty or not.