Removing (and replacing) asphalt base before tiling

[philimina]

We have a utililty room currently with lino flooring over an ashpalt base, which we would like to replace with slate tiles. The asphalt layer is broken and chipped in places (which could presumably be tiled over), but we need a little more clearance to lay the tiles under the existing doors into the room. It seems to come up fairly easily.

As the cellar below is damp, I'm worried that removing this asphalt layer will lead to damp rising into the room.

Any advice on how to proceed?
Would a replacement for the asphalt be recommended or should the natural tiles allow the floor to breathe?
Or are we better off leaving the asphalt as it and tiling over the top (dealing with the doors later)?