Remove tiles (and plywood underneath) around toilet without removing toilet
Anything is possible, but this falls under the Why On Earth Would You category. 

Removing and reinstalling a toilet is a fairly simple job for even a novice DIYer. Don't let that intimidate you into making your project far more difficult than it needs to be. 

Without knowing the details, here's a basic outline:


Close the water supply valve at the wall. 
Flush the toilet to (mostly) empty it. Wet/dry vac the remaining water out of both the bowl and the tank (if convenient). 
Detach the water supply at the underside of the tank. It's likely either a hex nut or a plastic wingnut. Be gentle. This stuff isn't cast iron. 
Remove the caps for the base mounting bolts (2 or 4), and then remove the nuts. 
Cut any caulk present against the floor with a utility knife. 
Wiggle and twist the toilet until it loosens, then carry it out. Optionally detach the tank for easier carrying. 


Reinstallation is basically the reverse, but there's the wax ring to deal with. Do a bit of reading here and elsewhere for tips on that.