I work closely with the immigration systems of UAE on the software development side. The following applies to you:

 - Once you get your new passport, then your passport number is changed; that means that your UAE visa is no longer valid.
 - There is no process in UAE immigration system to transfer the visit visa to a new passport. However, this can be done for residence/employment visas (not applicable to you).
 - You cannot get a new visa until the previous one is cancelled. Ask your visa sponsor to cancel the visa if you wish to obtain new visa before expiry of current visa.
 - Cancellation is not required if new visa is obtained after expiry of old visa.