If you experienced a fire in your home -- or perhaps your fireplace was not properly ventilated -- cleaning the resulting soot from tiled surfaces will be fairly easy compared to cleaning it from fabrics and other surfaces. Drying the soot first helps make its removal much simpler. Then with the help from a common household ingredient, you can clean any remaining soot from tiles and at the same time deodorize the tiles so they don't smell like smoke.
Open the windows if the soot is still wet. It will be easier to clean if it is dry. If it is humid outside, run a dehumidifier. If it is cold outside, turn on the heat.
Vacuum up as much of the dry soot up as possible with a vacuum cleaner. Use an attachment appropriate for the space. Wash the attachment pieces in warm soapy water when you're finished. This helps prevent the transfer of soot to other areas of your home when you use the attachment again.
Pour pure white vinegar in a small bucket and wring a rag out in it. Wipe down the tiles with the rag to remove the soot. Pay careful attention to the grout lines. Use an old toothbrush to scrub them clean.
Rinse the rag and toothbrush frequently in a second bucket filled with warm water. Replace the water as necessary. After rinsing, moisten the the rag or toothbrush with vinegar again and continue cleaning.