I'd like to slice and dice floor tile into pieces so I can arrange it in geometric patterns. I have CAD designs for the parts. Would any consumer grade CNC machine be capable of doing the job?
The answer is no!
Even if you ask: Is any professional CNC machine, build for metal/plastics/wood cutting, capable of cutting tile? The answer is still no! You can try it. It will work in the beginning but the machine will be destroyed soon.
Stone cannot be cut, except soapstone. Stone machining consist of crushing/cracking/grinding, cutting is impossible. Most common milling machines are build for cutting.
During the stone milling process there won't be any chips only "stone dust" There are parts of your machine which are able to protect all the moving parts of your machine from cutting chips. These parts have to be designed to provide protection against very small particles like dust. Usually they aren't, so the dust will come into your spindles, guides and gears and in the end it will damage them.
If you want to work with your milling machine for a long time then I would rather not recommend it.
You can do it. Then you definitely need water for the "cutting" process ( this will decrease the amount of dust) and very hard tools (diamond tools are recommended).