Vinyl and Linoleum Flooring - Should I remove the old tile?

I am about to install new peel and stick tile in two bathrooms. I already have the tile, the toilet is removed, the base board is removed and I have all the tools including a oscillating tool to make the job easier. I installed the original floor myself, which is peel and stick tile over an underlayment board. Should I remove the old tile? Or would it be a lot easier just to level with rock hard, sand the old floor and lay the new tiles on top of the old?

Welcome to the forums! Might I ask why you are replacing the original tiles? If it is from poor adhesion, then I would remove all the tiles. It is possible to retile over the existing ones as long as there aren't any dips, etc. I would offset my seams as much as possible so yours fall in the middle of the tiles below.

I used peel n stick in my kitchen and bath. Love the stuff. Lots of people have commented on it and thought it was rolled linoleum. I installed new 12x12 peel n stick over this ugly 18x18 peel n stick. It worked great. The stuff that was down that we were going over was solid and wasn't coming up, we just remodeled the bathroom and wanted new to match everything else. When installing I also added a dab of liquid nails to each corner and center of each tile. So far 2 yrs later, no problem. In the kitchen its been 3.5 yrs and still non have come up!
Good Luck

As chandler mentioned, I think the key is whether or not the underlying tile is secure. You never want to lay anything on top of a material that may pull up. If it's rock solid and you're just upgrading for whatever reason then you should be good to go in covering it up. Offsetting the seams is a must wherever possible.