As of 2014, the iPhone does not support Java applications, and Apple has never indicated any intention to support them, according to the iPhone FAQ. However, those who have jailbroken iPhones can install Java through Cydia, an alternative to the App Store.
While it's technically possible to install Java on iPhones, apps are unlikely to function well since Oracle, which owns Java, does not support the platform. Since Java is a virtual machine, it is difficult to port to new devices and it must be carefully configured to support the vast array of libraries that Java uses. Since Java is mostly used on enterprise devices and web severs, Apple has little incentive to offer support for its consumer devices.