Tile cannot be removed and reused. Installing tile requires a perfectly smooth surface.

If you particularly like some of your tile, what you might consider is carefully removing the tile you want to reuse and getting every possible bit of grout off the back. Instead of reinstalling it, use it to make a framed piece of art. A good framer can help you create a backing for the tile that will hold it in place and accommodate inconsistencies in the backing. It can then be framed and hung in the desired location.