If you want an RFID solution this thingy, http://www.farnell.com/datasheets/682634.pdf says it has less than 24 inch range and costs around $8 American, not including shipping. This is one of the readers from TI, http://www.farnell.com/datasheets/1719520.pdf, but I dont' think it has an antenna and that would be another expense. I wonder though if you couldn't have more than one antennae but use that same reader module. That way you could not only open the door but dispense a treat, daily food and/or water depending on where the dog, or the owner, brought a key fob.

The only reason I picked TI is that their stuff came up first during a search and I tried to get matching technologies. There were several different manufacturers but our company uses Newark for most of our common electronic parts supply so I'm basing cost off their prices. I'm not an engineer or a usually fully successful experimenter. I just had some time free for searching.

I'm assuming through all this you want to make sure your doggy door opens for your dog and not someone or something holding a magnet. Best of luck and please keep the forum updated.