Hi, Heather. Paint has to get down into the microscopic pores of a surface to stick. Glazed tiles have a smooth surface with no pores and are like a sheet of glass. So to get the paint to stick, you have to create pores on the tile. You can do this by sanding, usually with bad results, or chemically with agents that the homeowner should not be handling and at the high cost John mentioned. With sanding, you have to get all the glaze surface off and this is a very difficult and time consuming job. Any little area where the glaze remains is a potential flake in the future.
You might have an alternative if your tiles are collectible as some tiles are now. You could carefully remove the tiles and sell them to an antique dealer or architectural antique dealer and get enough money to partially defray the cost of new walls. You might start by taking a good, close-up picture of your tiles and checking with your local antique dealers or posting to the architectural antique dealers on the web to see if there is any interest.