Date: 4/27/2008 5:28 pm EDT
That black sticky adhesive is known as "cutback" used for many years for vinyl floor installations. It shouldn't be a concern in your case and it is impossible to remove all of it. Leave it alone after scraping what you can.
You CAN NOT install a subfloor material over the old tile you found there, that wouldn't be a good idea, no real way to attach it.
If your floor is within the recommended tolerances of any tile floor which is flat within 1/4" in 10 feet your Avaire tile installation should be fine.
Go for it and enjoy your new tile floor.