You should use COALESCE function http://www.postgresql.org/docs/current/static/functions-conditional.html
Try this:
SELECT
id,
COALESCE(CloseDate, ResolvedDate) AS FinalDate
FROM
cases
WHERE
(EXTRACT (month FROM COALESCE(CloseDate, ResolvedDate)) = EXTRACT(month FROM current_date)) AND
(EXTRACT (day from COALESCE(CloseDate, ResolvedDate)) = EXTRACT(day FROM current_date))
That should do it...