Possible asbestos danger
The short answer is that if you don't break the tiles, you are OK. There are plenty of resources available on the web to guide you in removing that flooring. It will be time-consuming, but not all that dangerous and not dangerous to anyone if you take simple precautions. Local regulations may vary so check locally.
My belief is that a decade from now, the EPA will be saying, "Oh, that residential asbestos thing, it was never really a serious problem." They have to wait for the people that over reacted to retire before they can say that.