Varies. In a situation with no construction/renovation, a permit may be issued in 7 days or less. However, depending on individual circumstances, the process could also take weeks to months. In cases with construction/renovation, health inspectors have up to 20 business days to review building plans.
A "New Construction/Major Remodel" Plan Check runs approximately $556. A Site Review can also be carried out ($281) in order to lay the groundwork for a smooth permitting process. (As of December 2010 - for current info see Fee Schedule (PDF) or contact the department.)
Review by the Health Department should be carefully integrated with the building permit/plan check process. Sometimes, plans are even submitted to the Health Department as a first step in this process. (Please note that while their processes may follow parallel tracks at times, a building permit and a health permit are separate permits.)