If its something other than exterior grade plywood or cement board, rip it out. What thickness of ply remains under it and what thickness are you placing over it? Are you putting cement board down also?

I found out what looked liked drywall on the floor is actually another layer of lineoleum covered by the backing of the layer that was removed . There is at least 3/4 inch spruce underneath, and we plan on installing another layer of 5/8 inch spruce.We then have a material called flexbond to put on to install the ceramic tiles.

Should we remove the layer of linoleum ? The installer I have told me we should leave it for support and soundproofing? Thanks very much for your time.

Linoleum offers no support and any soundproofing would be questionable. In fact if it is at all cushioned it is detrimental to a tile installation. I think he is handing you a line of bull so it makes his job easier. Tell him if doesn't want to do the tile job right to go into anothet line of work. We have more than our share of hacks in this industry. You certainly will have a stiffened floor by adding the 5/8" ply. Is it BCX grade or better. You DO NOT want to use CDX grade which is not rated for floor underlayment.

the store tells me the plywood i bought is BC480 which is good enough to lay ceramic tile on. Again thank you very much for your time.