There is a thermal bridge between the fabric and the wood that transfers moisture to the wood. Ruberoid is not an insulator. You need to eliminate the thermal bridge, either by creating a ventilation chamber between the two components or by installing an insulator between the fabric and the wood. There are different types of insulation such as polyurethane boards, etc., but a combination of insulation and a ventilation chamber and covering the fabric with tile or tile flooring to protect it from the sun and rain will be most appropriate.
Moisture infiltration will cause the wood to rot over time and destroy everything in the room.