Each county in Hawaii has its own regulations (or lack thereof) for zoning and permitting of vacation rentals. Please see the links below to get the most accurate information from each of the counties.
It may be of use to you to visit the planning department and planning commission web pages on each island on a regular basis to keep up to date with potential new rules and regulations that are in the works.
A summary of all ordinances on all islands is now available on the website of the Hawaii State Department of Commerce and Consumer Affairs. See http://cca.hawaii.gov/ins?s=Transient+Vacation+Rentals&type=usa for updated information posted on the department's website.
Here is a general brochure from the state on the Rental of Residential Property (current as of April 2016).
Property tax rates and a summary of ordinances for all the islands are (the last time we looked) on the Kauai website. It is useful for comparison and educational purposes. However, use caution because it is not known how up to date it is. If you are in doubt, call the county offices.
Hawaii (Big Island)
At the time of this page creation, the Big Island did not appear to have regulations regarding vacation rentals on its books. If you have or are planning to have a vacation rental on this island it is recommended you check back often to the county's website and pay attention to the Planning Department's postings.
Kauai has a permitting process in place and quite a bit of information on their website that is relatively easy to find. Start with this link.
Maui's website is a little difficult to navigate through. On their home page below is a little box on the left hand side with a binoculars picture. This is where you can input your searches. Start here with the website and then input "Transient Vacation Rentals" into the search box choosing "What is a Transient Vacation Rental?" You will find definitions and answers to frequently asked questions.
See this link for Homes compared to condos: http://www.co.maui.hi.us/1874/Short-Term-Rental-Home-STRH-Application
Oahu (City and County of Honolulu)
This website has information scattered all over it but it does not appear to have a central location or locations for Vacation Rentals as Maui and Kauai do. There is a clear search area but results are not as clear as on the other islands. There does not appear to be one place to get answers on vacation rentals on the website. You will have to do a lot of digging.