By now there's been countless amount of rumors for larger screened iPhones. For example here's what iPhone 6 is supposed to be coming with. Most rumor postings only focus on the physical size of the device, which sure is important, but pay very little attention to the amount of pixels the screen would have.
The reason why the pixels matter is that Apple has been very conscious to avoid fragmentation in a device family, as much as it can be avoided. So when going to high-density "retina" displays, the pixels were scaled exactly 2x on both iPhones and iPads, so that old apps would run as-is without getting screwed by the aspect ratio (stayed the same) or pixel scaling (2x is as nice as it can get).
So when it comes to screen sizes and pixels on those screens, the real question is, how can you make the new "large" screens behave well in the non-retina + retina family?
The answer from the pixel side of things is relatively simple: have a scaling factor that is preferably whole integer (no decimals). Now, considering that iPad Mini (7.8" screen) sets a spiritual upper limit to the screen size, we are talking about screens with physical size of 4-7 inches diagonal. Within that range, doubling the pixels of retina screen (640px width => 1280px width) gets us into extreme dpi range, up to 600+ dpi. Those kinds of screens are fairly pointless to human eye (retina, i.e. ~326dpi is the sweet spot for a phone) – as well as very hard and costly to manufacture. On the other hand, Apple is probably not interested in going below retina resolutions, so 326dpi is the lower limit.
So 2 times the retina pixels is overkill. But there are no other integer-based scaling factors, right? Well, there is one: 1,5 times the retina. Because it is 3x the non-retina screen. The iPhone development environment continues to calculate sizes and positions in the non-retina scale, so introducing a "3x" scale as a sibling to "2x" scale is fairly trivial change. Of course the scaling of the pixels isn't perfect for old apps, but then again, retina (2x) has already very high pixel density, so with a even higher-dpi screen, the blurriness could be fairly small effect.
With 3x scaling, and iPhone 5/5S aspect ratio, we get 960px by 1704px pixels screen. At 326dpi (same as iPhone 5), that would translate to 6,0 inch screen, which could be a bit too large for Apple's tastes.
The iPhone 6 article linked above, speaks about 4,7" and 5,5" screens. Let's look at those sizes then.
The 4,7" would be the highest density display at 420dpi (again, 960px*1704px). While it's a high dpi, it still sounds doable. From user experience point of view, it would be merely 16% physically wider screen than the iPhone 5S, and would feel a lot like iPhone 5S, albeit with much more (non-discernible) pixels under the hood.
The 5,5" would have 355dpi screen, which is not much different dpi from the iPhone 5S. The device would feel much bigger though, nearly 40% physically wider and taller than iPhone 5S.
With the 4,7" and 5,5" phones, the existing apps would run just fine. Thanks to the extreme 420dpi resolution on the 4,7", the blurring from scaling the pixels of the old apps would be barely visible. On the 5,5" screen it would probably have small, but noticeable-if-you-look-for-it, effect.
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All the discussion above, assumes that Apple is interested in keeping the contents on the screen exactly the same across the devices in the family (i.e. all the iPhones in this case). With larger screens it's of course possible to start re-layouting the content, for example putting more icons to the home screen and so forth.
However, at launch, iPad mini introduced a new physical screen size to the iPad family, and yet Apple kept layout the same as in original iPad. I foresee Apple continuing to use this pattern in iPhone-family too.
