Generally speaking, tile is usually installed over two types of foundations. Concrete slab and wood sub-floor are most common. It’s important to understand that when tile repairs are needed over a concrete floor, You may be able to remove the tile and the existing vinyl to get down to the concrete. However, if the vinyl was not glue or secured 100% to the concrete, then vinyl removal is needed without disturbing the surrounding tiles.
Why do you need to remove the vinyl? Because if the vinyl was not secured to the floor, it does not provide the proper substrate foundation for tile installation. Think of the vinyl as “floating” on top creating a improper substrate for tile installation. If the tile was installed over the vinyl, it could possibly fail based on any movement between the concrete/vinyl. Tile substrates need to be solid and secure. Any slight movement causes excessive stress on the grout which can eventually start to fail and develop cracks surrounding the tiles that are not bonded correctly.
Another consideration is when tile is installed over vinyl on a wood sub-floor. If you have tile bonding issues with this configuration, it is suggested to completely remove everything down to the original sub-floor which includes vinyl and it’s underlayment. The reason why is because when vinyl is installed over sub-floors, it generally has it’s own underlayment consisting of particle board or compressed chip board. Either are not proper substrates for tile due to their excessive expansion/contraction from moisture/water. You may be able to repair 1 or 2 tiles but realize that this is not the professional method and could fail in a short period of time.