Toilet Ring, Seal, New Tile Floor

That's just a compression flange. Loosen the 3 screws inside the ring, then remove the screws that secure the flange to the floor and you'll have it off in about 2 minutes.

I'm positive I can remove it, but I'm question whether I'd be creating other problems. The flange is fastened to the concrete under the tile. Rather than removing and having to re-fasten to the concrete I'm thinking to leave it alone, since it is already well secured to the concrete. I have limited tools / knowledge since I'm a DIY'er. Sorry I should have described it is connected to concrete & tile. I would hate to have a problem re-fastening to the concrete.

Not going to be an issue, just remove the ring.

You'd need even more sophisticated tools to cut-out such a neat circle around the existing flange! If you just try to carefully break it with a hammer and chisel, then you'll most likely cause the screws to lose their grip anyway. All you need to re-screw back into the concrete is (maybe) a hammer drill and a screwdriver.

Another problem with what you're suggesting is that if there's a height difference (and there likely will be) between the old tile that the flange is sitting on and the new tile surrounding it, then your toilet could rock and eventually start leaking from the (broken) wax seal.

The flange has 5 concrete rivets on the outer part. I can't just unscrew the fasteners. The only screws are the 3 allen head screws on the inner bore. I'll have to pry up the rivets. I did find the part on the web, its a Bruco GH950P 4" PVC compression flange, either that or it is a 3" of the same type.

In that case, drill or grind the heads of the rivets off, then pull the shanks out after you've removed the flange.