And the really big problem is that a varying aperture doesn't work well with our sliding shutter curtains - we don't expose the full image at the same time, so a varying aperture while the shutter curtains moves would result in the top and bottom of the image to be taken with different aperture. And that would require the shutter curtains to also move with different speeds so they move quick when the lens is fully open and move slowly when the lens is stopped down.
This is the major problem. You end up with an open slit traveling across the sensor and as it does, you have different apertures running. So you would get decreasing exposure as the shutter traveled and end up with a light to dark image. Assuming you have some kind of computer control that could compensate by varying the time that each aperture was open to give an even exposure, you would have a depth of field that expands as the shutter travels down the image.
On longer exposures, the effect would be different. Assuming the shutter opens completely and then the aperture makes its run, then you might be getting somethign interesting. You would need to do some calculating but it would be straighforward. As others mentioned, a manual camera with a manual lens on film would be the easiest way to do it and see what you get. I would guess that the result would be basically a multiple exposure image with partial exposures combining to add up to the proper exposure. I also guess that taking multiple images and blending them in post would product the same result. So for example, you have 5 images at f2.8, 4, 5.6, 8, and 11 and you do all 5 as layers in photoshop with 20% opacity.
But hey, we're all speculating. Part of the fun in photography is the experimentation. Get out there and do it!