I think this is a complicated topic. If we have a non-circular pinhole, the diffraction pattern will change. Instead of a uniform Airy disk, we get and "Airy square" or "Airy ellipse".
I've just spent the past couple hours looking at diffraction patterns for different aperture shapes... here are a couple neat links I stumbled across ( one for microscopy and another for acoustics ):
But the image formed on our film or paper is a complicated overlay of the diffraction patterns of all the points on the image. For me at least it is not obvious what the overall effect will be from overlaying this "point' pattern. Now I'm more curious than ever about the effect on perceived sharpness or other more subtle effects ( like the characteristic "glow" of highlights in a zone plate photograph. )
After thinking more, I don't think my microwave analogy above works at all. Those are waveguide properties, when the aperture less than the wavelength. Pinholes are huge compared to that and it's diffraction around the edges that we're talking about here, which does happen in microwaves ( around the edges of buildings for example ) but is not important at the edges of a slot antenna.
Non-circular apertures are used in various applications like astronomy and microscopy. I think it might be fun to experiment with a more "qualitative" and artistic goal in mind.
Also an aside: I'm getting sucked in to making things too technical, which I have a strong tendency to do. The reasons I like pinhole photography are almost the opposite of "technical"... it's the magic of the light landing directly onto a piece of paper in a coffee can making a wonderful photograph! I need to be careful not to suck all the joy out of it and making it into "work" instead of play. I have no fear of differential equations but my next step needs to be to make something and play with it instead of trying to solve this with 2-d convolution or whatever.
AgX has a good idea and a simple one that would be easy to mount. Also it will not be hard to construct two rotating slit plates from razor blades, so that the aperture could be adjustable parallelograms... Time to stop typing and start doing!