Ceramic tile is a beautiful, durable surface that can last for a very long time. Over time, it may become necessary to remove a portion of tile that is already installed, while preserving the remaining tile. This can be attempted by a do-it-yourselfer, although the results aren't guaranteed. Even experts can crack a tile or make an error in their cut.
Remove everything in the area where you will be working. Tape and mask off your room from all others in the house. You must have good ventilation, plus a fan. Crystalline silica dust can kill you. Make sure there are no children or pets in the house.
Use a grease pencil and straight edge to mark your line well. Once you start, it will get very dusty and hard to see. Tape along the "keep" side of your line to help protect the edges from chipping.
Put on protective gear and place your fan where it can move the room's air toward the window. Have a helper stand to the side of your work area, ready to vacuum with one hand and squirt water on the cut site--if it is horizontal. You will need to work as a team, one cutting, one reducing the airborne silicate by vacuuming immediately and squirting water to assist in the cut. Water will reduce dust. If your cut is on a wall, use a spray bottle.
Use an angle grinder to cut the tile. If you are cutting a curve, make small, straight cuts and return later with a ceramic tile file or your Dremel tool to clean up the curve.
When you get near the wall, your grinder will be too big. Use your Dremel's tile-cutting bit to cut in the narrow spaces. Keep the vacuum going, but not the water; your Dremel isn't design for wet applications. Follow Dremel's instructions for using their bit. Keep all electrical cords away from the water.
Vacuum your entire room with the shop vac before removing your respirator. Keep your fan running after you think the dust is gone, because it probably isn't. Wipe down your room with a wet rag and then leave the room. Take off your respirator outside. Try not to track silicate dust everywhere. Get out and strip down.
Wear long sleeves, trousers and gloves when cutting tile. Place your dirty clothing in a sealed bag and launder them at a Laundromat. Hose off your boots outside. Keep everyone out of the room for a good 12 hours. Wearing a simple face mask, scrub down your room to remove whatever is left of the silicate.
Make sure your respirator is rated for crystallised silicates.