No, you do not need to unlock all the islands in the game to be able to use this mod. However, you are encouraged to at least try playing through the regular version of the game so that you know how it works.
Then, once you have gained some knowledge about how things work, install and try out the Dragon Mania Legends Mod Apk. This will allow you to experience what it feels like without having too much wasted time or effort invested in it if it does end up causing any problems with your phone or device.