A wood saw will cut pvc fine as would a hack saw, sawsall, cable saw (made for this), and even some string. You'll need to cut it off low enough to put in a coupler (basically a tube with a ridge in the middle so you center the two pieces when you glue it to the pipes) and still have enough new pipe to stick the flange onto.You'll also need to keep the hole around the pipe big enough to get the flange to sit down over the pipe and sit flat on the floor. The flange fits on the outside of the 3" pipe and flares out, so its diameter is fairly large compared to just the pipe sticking out of the floor. That dimmension varies by the flange you choose.
If you notch the tile (much easier than trying to drill through them later) when you install them, you can then screw the new flange down into the subflooring. Some porcelain really beg for diamond drill bits if you decide to try to drill rather than notch.
When you are finished, the new flange should sit flat on the top of the new tile, and be firmly fastened to the floor with no gaps under the rim to the floor.