Speaking first on a tile laying perspective rather than plumbing concerns. I always remove whatever I can from the walls or floors I'm preparing to tile for a customer. That being sinks, toilets, mirrors, etc... I'm looking out for my customer's future changing tastes of new bathroom goodies. And those goods come in all shapes and sizes. You may end up with a bare spot on the floor or wall if you decide to change any fixtures in the future. You may change from a cabinet sink to pedalstal or in your case maybe a different toilet in which the base of your old is wider or longer than a new future selection. If so, trying to find the same tile to cover up the new empty spots on the concrete floor or Durocked wall can be painfully impossible. The tile could be out of stock or production from the manufacturer. You may have problems finding the same grout hue as well. Now wood floors can be tricky sometimes depending on how thick the new tile is verses the original flooring. But a little shimming under the toilet flang usually does the trick. From a plumbing stand put, I leave it up to my local plumbers to replace any and all fixtures. They tell me what they need and I abide. I get most of my business from seven different plumbers in my area. As long as I keep my floor tile level, I don't get any complaints. Pull that toilet.