While Santa receives most letters addressed to him at the North Pole, some are made available to organizations and individuals who want help those in need. Many Santa letters will be answered by a variety of Santa's helpers — charitable organizations, employees of local Post Offices and individual volunteers who want to help make children's holiday wishes come true. Contact your local Postmaster to find out how and where you can participate.
Santa’s Reply Letters – North Pole Postmark
The Postal Service is helping create holiday memories for children. Parents, guardians and other Santa’s helpers can craft a response from Santa and mail it in a second envelope to the Fairbanks, AK, Post Office. Postal elves at the nearby North Pole Post Office will open the envelopes, postmark Santa's reply letters and mail them back to the children they are addressed to.
From what they put in their FAQ, most letters "to Santa" are actually sent to a local family proxy (i.e. aunts and uncles, etc). However, if you want to send something and get a "return letter from Santa", they have a courtesy service to postmark letters from "The North Pole".
No answer is given on what they do with letters that kids send out "To Santa" with no other address.