Instragram recently made their API public, and a service has been born of this generally accepted to be called "Instaprint" (coined by Engadget, it seems).
Instaprint is essentially a modified Zink photo printer with internet connectivity and tie-ins to the Instagram API, that automatically prints items tagged with any hashtag you choose. For the time being, the company is renting and demoing these "Instaprint" boxes at SXSWi.
See prints being printed live on the embedded video on
Engadget