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Our house had asbestos wrap on the furnace and asbestos tile on the entire second floor, all of which we knew about when we bought our house (which was a few years before we had kids).

From our research and talking to asbestos abatement people, we came to the following conclusions:
-The stuff on the furnace was the highly friable kind, meaning that it can easily become airborn if disturbed. Our furnace was original to the (1900) house and we wanted to update, so we had the furnace asbestos professionally abated. It was $10,000 and we had no furnace at the end But it was a cost we knew going in.
-The asbestos in our kind of floor tile was well-encapsulated and hence not very friable. My husband did a big search of the research literature and was able to only turn up one study on this type of asbestos, but the study indicated little risk from removal if proper measures were taken. Our abatement people encouraged us to do this job ourselves (legal in our state) and walked us through the abatement measures they would take for the floor tile (for example, seal the room; negative air pressure; keep everything wet; protective clothing; thorough clean-up, etc) . We did it ourselves and saved about $5000.
-Our neighbors had the same tile and covered it with wall-to-wall carpet.

Edited to add: If the tile is in good shape, I think it poses little risk if left in place. From what we read, the traditional asbestos vinyl tile needs to be pulverized to release the asbestos.

Other than my random bouts of anxiety--which I get about everything--I am totally happy with our choice. It was stressful at the time, but I think we did a good job with the work and I'm glad to have the material just gone from the house. The floors underneath weren't what we expected--painted wood--but we ultimately decided to repaint them and are pretty happy with them. (They are white, and our second floor feels like the middle of a bright, happy cloud on sunny days :))

As I said, though, we did it before kids. That made it possible to do ourselves and probably kept my anxiety levels lower.

We had our very nice elderly neighbor accidentally sand out asbestos tile floors in our new home (we had told him not to touch it and just lay over, but he didn't think light sanding was a problem.) we had the tile tested and then the air. (Tile was positive, air negative). It cost us $1000, but they determined their wasn't any asbestos in the air, so it was worth it. What i learned was they still sell asbestos stuff (except insulation). So potentially anything could be hazardous, even stuff now. And it's hard to tell on site (though there is a database). We covered with linoleum (it was only in bathrooms) and are happy with the choice. There was also some in the kitchen, and we put hardwood over it.

Also wanted to add: our tile was installed over large sheets of plywood. This allowed us to take up entire sheets of tile/plywood at one time, without disturbing the glue bond or breaking many tile. If the installation of the tile had been different, I don't know that we would have removed the tile ourselves.