The iPad's 4:3 aspect ratio is unlikely to come to the iPhone. The iPhone's form factor would have to change considerably, to a more squarish shape. A 4-inch screen with a 4:3 aspect ratio would have dimensions of 3.2 x 2.4 inches, or about 0.09 inches wider than the iPhone 4's overall width. The one advantage of going this route: a 1024 x 768 resolution would still have the iPhone's display at a Retina Display level of quality, and it would also simplify producing "universal" apps between the iPhone and iPad. With both devices at the same resolution, developers wouldn't have to change what they're doing at all; they could go on making one version of apps for the iPhone 4 and below, and another version for the iPad/iPhone 5.
The main problem I see with a 4:3 aspect ratio iPhone is that it might be too wide to hold comfortably as a phone. Tacking another half inch or so onto the current iPhone's width might not seem like a big deal, but some folks with small- to medium-sized hands already complain about the iPhone's "bulk." A big, square iPhone might not look all that stylish, either; lots of people seemed to hate the third-gen "fat iPod nano," and people might react with similar distaste to a "fat iPhone."
If Apple really is going for a 4-inch screen in the iPhone 5, it could conceivably use either a 16:10 or 16:9 aspect ratio without having to change the overall handset dimensions at all, though in both cases the home button, front-facing camera, ear speaker and sensors would all need to be shifted toward the edges of the device. The handset might even have to increase in length to make room for all the internal hardware, though it could remain at about the same width.
Apple currently has 16:10 displays in the MacBook and MacBook Pro lines, so it's not inconceivable for this aspect ratio to show up in the iPhone 5. At 4 inches and 16:10, the screen dimensions are 3.39 x 2.12 inches. Looking at the width, this leaves 0.095 inches (about 2.4 millimeters) between the edge of the screen and the edge of the phone itself -- very cramped, but possible. If Apple wants to keep the screen vertically centered (and it's all but certain that it does), the screen would expand by 0.315 inches (8 mm) in both directions. This would put the screen edges roughly where the ear speaker and the centerline of the home button are on the iPhone 4, meaning all those components would need to move. Additionally, the home button would have to shrink by a fairly substantial amount in order to fit; it's currently about 11 mm in diameter, and moving it downward by 8 mm wouldn't leave enough room for the currently-sized button.
At a 16:10 aspect ratio, Apple could keep the iPhone 4's display at 640 pixels in the portrait dimension without substantially sacrificing resolution -- it would be around 300 ppi instead of 326, which isn't much of a loss. The resolution would change to 1024 x 640. This means apps formatted for the iPhone's current 3:2 form factor would run "pillarboxed" with 32 blank pixels on either side -- at roughly 300 ppi, that works out to about 2.5 mm of blank screen on either side -- unless developers decided to program the new resolution into their apps.
The 16:9 aspect ratio is far more common than 16:10, with many films and almost all widescreen TV formatted for it. A 4-inch, 16:9 screen would be 3.49 x 1.96 inches. This is only about 1.5 mm wider than the iPhone 4's screen, and definitely able to fit in the current form factor's width. The length is another story -- it's a full 0.1 inches (2.5 millimeters) longer than the 16:10 screen, meaning the home button and other components would have to be shifted even farther toward the edges.
The pixel dimensions are slightly more problematic for this aspect ratio from a developer's perspective. The portrait dimension could remain at 640 pixels with a negligible reduction in PPI, but the landscape dimension would change to 1140 pixels (1139 technically, but odd numbers and resolutions don't mix well). 1140 x 640 is a pretty oddball resolution though, and current 3:2 formatted apps would run with 90 blank pixels on either side -- roughly 7 mm of "pillars" on either side of an app. Not particularly appealing to the eye, but then again, supporting yet another resolution may be asking too much from smaller developers.
No matter which way you look at it, changing the iPhone's screen size and dimensions comes with tradeoffs. Simply bumping up the screen size without changing the aspect ratio or pixel count means the overall resolution drops, possibly below "Retina Display" quality -- but changing the resolution comes with its own difficulties. Alternate aspect ratios aside from the iPhone's current 3:2 format have some advantages, but in each case they also present their own challenges.
Given that most people seem perfectly satisfied with the iPhone's current screen dimensions and aspect ratio, the easiest course of action, and seemingly the most likely, is to leave the iPhone's screen as-is for now. Either way, we won't know for certain what's going on with the iPhone 5's screen for another few months unless there are more leaks in the meantime.
Update: You asked, so we delivered. In the gallery below, you'll find mockup designs featuring all of the aspect ratios discussed above. Some of them look a lot better than others.