I am going with Dodgeman about secure and sound back walls. But.
I always in my job, removed the tile. Its soo easy IF its mastic glued to drywall, either green or cementious. If its mortared to lathe, then its a real headache and I would NOT by any means remove it because TNT will not budge that.
As one stated, line the bath tub and once you remove one tile then an old chisel will take them off faster than you can ever imagine.
Here is my reasoning on removal: If you attached them to the existing walls, I would want them staggered over the old tile grout lines. In other words NOT to line the new tile directly over the old tile.
Homes and walls do move and flex. Tile, ceramic, will crack on the grout lines usually and those are very easy to repair. But if the wall flexes with two staggered layers, the crack will not follow the tile grout lines and end up cracking tile.
Then you also have the additional 1/4 inch or more increase in the wall depth. Can all fixtures, and stuff handle this easily?
Tough call, both answers have due merit and not debatable really. The key is a good backing.
Now, on the tile of formica, that also depends on a good fixed formica backing. Any bubbling, and lifting, then the formi will not work, But if down sound, then rough it up as stated and go for it.
On requests like this when there are two ideas, its difficult to jump in and play ref, because one really needs to see the site and anaylize the situation. We cannot do this here, all we can do is to give our thoughts, based on our experiences and our problems in doing this and our thoughts about consequenses of doing this.