How would applications on the iPhone adapt to a bigger iPhone screen?
If Apple were to increase the screen size, it would likely be done in the same fashion by which they have done it each previous time - in such a way that apps, while maybe not looking as great as they could, are functional and visibly proportional.

When they introduced the Retina screen on iPhone 4, they kept the screen size the same but doubled the resolution. So while pre-Retina apps might have looked a bit blurry, they were still functional and had the exact same proportions.

This same process occurred when the Retina screen was applied to the iPad 3.

When the 4" screen was introduced on iPhone 5, they kept the same width but increased the height of the screen. Functionally this meant that all apps operated identically, with the sole difference being black letterboxing at the top and bottom of the screen. But since the pixel density did not change, nor the width, apps looked exactly the same with no proportion issues or visual degradation.

Apple has shown themselves to be very thoughtful about making such changes and there is no reason to believe they wouldn't be going forward.