First test the existing floor for asbestos. You can buy a test kit from your local hardware store. Included in the kit are instructions on how to take a sample and then where you can send the sample to a lab.
- If your existing flooring tests negative you are good to go.
- But if not realize that improper removal and disposal of asbestos laced product can be both a hazard to your health as well as your wallet (think fines).
Assuming it tests negative, you will need to install an underlayment before a new floor. Most flooring products require a clean smooth 1/4"-3/4" under layment to minimize the opportunity for telegraphing of chips, cracks and other blemishes in the the existing floor. Consider how this added height affects door thresholds before you start your project.
As to covering an existing floor that contains asbestos with an underlayment, most jurisdictions (check yours) will not allow you to screw/nail down into any existing flooring that tests positive for asbestos due to the concern over disturbing the asbestos fibers and making them air born and therefore a health hazard.
For more information:
Read "What is the best way to seal/encapsulate vinyl asbestos tile?" a Q&A answered by Joel Hirshberg.