Linux software engineer Dave Hunt created the Pi-Rex as an educational project (and, well, to end his dog-induced sleep deprivation). The project uses a noise detector wired to the Raspberry Pi and a weight/pulley system that swings the door open.
It’s not an entirely practical project unless you’re OK with leaving your door open after the dog has opened it. Also, it’s probably not a great idea if your neighbours have barking dogs or your area has other wildlife (or thieves) that would like to come in for a visit. Still, it’s a system that you can build upon or that might inspire similar creative projects.