i recently bought a house built in the 50's. The living room and hall were covered with old carpet, but i knew from being in the basement that there were hardwood floors underneath. My plan was to refinsh the hardwood. Upon removing the rug, i found very old 9" vinyl or linoleum tiles which I believe contain asbestos. The tiles are not broken. I don't want to do anything that would damage the tiles releasing the asbestos fibers. And removing the tiles is definitely out of my budget. Can a floating hardwood floor be installed on top of the tiles? Do I need to cover the tiles with something else first?

Yes, you can float a floor over those old tiles. If any are "crunchy", you'll want to remove the broken pieces. The only thing that will need to go over the tiles is the foam for the floating floor. Once covered, it is harmless unless disturbed.