I have run Aperture on a Mac Mini without a screen on both OS X 10.7 and 10.8. However over screen sharing the image quality is not what it can be when running Aperture directly on your computer.
I also shared my Aperture library from the Mac Mini, but it gives (as indicated by Apple) a lot of instability.
You're right William. With no screen plugged in Aperture will not launch. So I ended up buying the smallest cheapest HDMI TV I could find and when that is switched on the Mini runs Aperture just fine. And since I got one of the quad core i7 Minis it runs screamingly fast too - particularly as I have my Library on a Thunderbolt RAID drive. But often I don't bother with firing up Aperture on the Mini - I run it on the iMac and access the Aperture Library remotely - it works ok wirelessly but now that I've put in a gigabit Ethernet switch it is as good as if the library was local. I find that configuration to be rock solid for anything that doesn't need the raw cpu power of the Mini (batch processing with noise ninja for instance).