Carpet tape can be removed from tile floor by gently scraping, starting at the edges and working toward the center. Depending on the type of tile and its durability, you need to be careful not to scrape the tile.
A wooden scraper, or a flat piece of wood, such as a paint stirrer, may do the job. If the tile is a durable stone and you can work carefully, a razor blade works well.
Work slowly and carefully, peeling away pieces with your fingers as they come loose from the tile. You may need to use soap and water, or window cleaner to remove any remaining residue after scraping.
Some tile floors will not be harmed by chemical removers, but some will. Always test in an inconspicuous spot before using a chemical remover.
If the tile will handle it, you can use a commercially made product, like Goof Off or Goo Gone, but you might also be able to use something you already have around the house, like fabric softener mixed half and half with water and sprayed on the carpet tape. You can also use rubbing alcohol, window cleaner, or in cases where only the sticky residue remains, peanut butter.
Be sure and thoroughly clean the tile with a mild soap, or tile cleaner, and water after using any chemicals.
If you have delicate tile and scraping or chemicals could cause damage, a little hard work will remove the carpet tape. Start by loosening it as much as you can with your fingers, starting at the edges and working in.
Then, dip a rag in very hot water and lay it on the carpet tape. Allow it to sit for several minutes.
Then wring the rag out and begin scrubbing back and forth with the rag, picking pieces away with your fingers as they become loose. When most of the tape is gone and only the sticky residue remains, add some of your regular tile cleaner or soap that won't damage the tile to the water, dip the rag and begin scrubbing again.
Keep scrubbing and removing loosened pieces until the carpet tape and the residue is gone.