If someone is hired to a job that is classified as part time, and over the course of the next three years, management (public sector) decides that the job should be full time, can they fire that person who has been in the job if they do not want to take on the full time committment? Or, must they make some concessions to allow that person to stay on in the capacity in which they were originally hired?

Since you say public sector, is there a CBA involved?

Barring a contract that says otherwise, if the employer decides that the job needs to be full time and the person in the job cannot or will not work full time, then yes they can be fired and a person who CAN work full time brought in.