I have a feeling that the tile surround is already partly covered by your mantle, unless the tile was installed precisely within the opening of the mantle, which is unlikely. It would probably be optimal to remove the mantle so you expose all of the tile surround, and install the granite on the tile and then re-install the mantle.

Can the mantle be removed without damage and re-installed? It depends. If it's made of real wood and not some kind of particle board that they sometimes use these days, it can probably be removed without too much damage. The other concern about damage is that the mantle is probably covering some of the tile surround and, beyond the tile surround, some sheetrock. That means it may have been glued in place. You might therefore have to contend with damage to the sheetrock before re-installing the mantle.

Another problem you may have, which you already mentioned, is that the granite will add "thickness." That means that when you re-install the mantle, it will be about an inch farther into the room than before. You will need to figure out some kind of adjustment for that.