Soften the grout around the tile. Spray a mixture of equal parts water and basic household cleanser on the grout around the tile, and work it in with a toothbrush. Allow the mixture to sit on the grout for about an hour.
Cover the surrounding tile with painter's tape to protect it from damage.
Scrape a sharp utility knife on the grout around the tile. Continuously scrape the blade of the knife over the grout to loosen and remove it from around the tile.
Insert a putty knife in the grout line underneath the cup holder. Apply pressure to the putty knife, and slightly push up on it, pulling the cup holder away from the wall.
Insert the putty knife around the grout lines on the sides and top of the cup holder. Apply pressure to the putty knife, and slightly push in toward the tile while doing so to further pull the cup holder away from the wall.
Gently move the cup holder back and forth and up and down to remove it from the wall.