Have it tested. It's not terribly expensive. Removal and disposal is.

First things first: how old are the "layers"? Asbestos products were no longer available after the mid/late 80's. And traditionally old linoleum (real linoleum NOT sheet vinyl that people say is 'lino') was not installed using cut-back adhesive (the stuff that contained asbestos).

As einportlandor suggests, it is best to have all three layers tested (the gray stuff, the tile and the black adhesive). You will need a sample of each one (three samples in total) sent to a testing facility. It will cost $50 - $100 for each item to be tested (could total $300 for all three samples). Once you know, you can then figure out how to go about this renovation.

Of course the easiest thing would have been to add a layer of plywood over the old sheet product and then installed the new product over the plywood. But I think that ship has sailed.

We tested the tile and adhesive in our basement and found that the tile was fine, but the cutback adhesive had enough asbestos to make it hazardous. I considered removing it myself, since liquid removers do not release asbestos into the air, but finally ended up paying an abatement company. They used some type of liquid remover (I could not see what it was because I did not have the protective clothing), and then sucked it up with a water vacuum. When they were done the concrete had a kind of dirty gray cast, like you get when you try to remove oil stains from concrete. However, it took floor paint very well and provided a very even finish.

I had about 450-500 sq ft of concrete cleaned up, and IIRC it was about $1,500. Its certainly not a small amount of money, but if you think that will break your renovation budget, just wait until you get the electrician and plumber bills.

Bruce