Dubai and Bur Dubai and any area of the city does not have a ZIP code as the others Middle East countries. In case you have to fill a form onlne with the zip code of Dubai, just type Dubai and it will be enough.
In order to have a address to receive mail in Dubai and UAE, you need to purchase a PO Box in a Dubai post office. You can open a PO Box with your company details (if you own a company) or also privately as an individual, the cost is around 600AED/year. There is no private residence posting delivery unless you use a private courier like DHL or aramex so the use of a PO Box is mandatory to have a permanent physical place where to receive your mails.