Would towels under floorboards (for sound proofing) be a fire risk?
There are two kinds of sound problems: 1) airborne sound, and 2) impact sound. 


Airborne sound is talking, music, etc. If you’re trying to stop this kind of sound, then yes, additional material that absorbs sound will help. 
Impact sound is sound that is generated by footsteps, tapping on floor, etc. This kind of sound travels through the flooring material and reverberates into the space below. Towels will not interrupt this transmission. Rather, one surface needs to be isolated from another to stop the transmission. That is to say, the upper flooring and structural components need to be isolated from the lower ceiling and its structural components. 


Towels are not the best material to solve either sound problem. In addition, cloth towels have a very high rating for products of combustion and contribution of smoke. 

Every material is rated for fire and smoke. Cloth is rated very high for both. However, in single family residences it is not prohibited.