[Zeke]

Asbestos is only harmful when it's disturbed. The only real danger there is if you sand the old floor to apply a new one. Chipping and scraping the floor out is relatively safe. If you have any concerns beyond that, you can keep things slight wet while working. Vac the area with a HEPA vac (again because you are making any residual AB airborne) and put the new floor down. There will be a trace of AB in the mastic, but it's fully impregnated. Just use common sense.

Again, the only stupid thing you can do is sand the existing tiles or the bare sub floor after removal. Even then, there won't be much AB in the base. Breaking tiles releases a infinitesimal amount of AB which can be contained by vacuuming, not sweeping, with the option of using some water to contain any dust you might make. Put a little detergent in the water as a wetting agent to keep it from balling up on any wax.