I was asked a question recently that I think crosses the mind of many home renovators. Can I lay my stone tiles on top of the linoleum that is already in my kitchen?
This is an interesting question, and the answers often vary. From my perspective laying tile of any sort over an existing floor is not the best practices kind of thing to do. Think about it like this. If you are laying your tile over an existing tile or over linoleum, how secure is your current flooring material. This is an important issue because basically you are thinking about adhering the new, and likely more expensive floor to it. If the bond between the sub-floor and the linoleum is not strong, there is potential for movement and as a result breakage/cracking of the new floor. Is it worth the risk? To me it is not.
If you decide this is the way you are going to go. Test the bond of the old tile. Lightly tap it and see how secure it is. If it is loose at all, its a bad idea. If the bond is strong, next test it in a corner to see how the thinset adheres to the current tiles. Also note, that there may be issues with the height of the new floor. Because you are simply laying another level on top, there may be areas around doors, cabinets etc. that may look strange.
If it is really just virtually impossible to remove the previous flooring try using a cement board on top of it. At least this will give you a membrane of sorts to eliminate some of the risk associated with a loose sub-floor.
Food for thought anyhow.