Sometimes after tile installation and even some months later as temperatures change, saltillo tiles can have a divot like piece that will just pop out. It's most often caused by a pebble getting mixed into the clay and it causes a weak bond. I have tiles with small chunks missing that I feel just add to the character of saltillo, but I ended up having a big piece of a tile pop out right where the the kitchen chairs are and it was causing the chair to wobble, so I had to glue the piece back into the tile. It didn't add character, it just looked like a cheap fix. So we took it out and replace it with a new tile.
The only was that I know of for removing a tile is to break it out, or chip away at it until it's gone. It is a very dusty project, so we made a tent over the tile.
It ended up coming away from the grout pretty cleanly, so we left the grout and found a tile to fit.
Mixed up a small batch of glue and grout for the new tile.
And after it was all set, I sealed it.