If you setup Openfire to run as a service, then you don't have to login to any account, or someone can be logged with his account and doing his work and service will be running along in the background. Openfire: Installation Guide
Not sure about the Raspberry PI. You would need some operating system on it, presumably some minimal Linux OS without graphical interface, so this can get too tricky.
The hardest part would be to make it work on the Internet. You will need to make your server visible from the Internet, by opening ports in your router/firewall, possibly setting up DNS name if you want outside accessing the server by its name and not the IP address of your router.