Once again, this is false. I am currently serving at a USSOCOM unit with qualified Jumpmasters and a large number of both SOF and regular Airborne troops (multi services). There is no mention of knife sheath materials in the USASOC REG 350-2, Airborne Operations manual. In fact, the only place that a fixed blade knife is mentioned is under individual equipment:
"(c) A strobe light (night operations only) or service approved day/night dive flare and a sharpened, fixed-blade, sheath knife. The knife will be secured in its sheath to prevent loss during operations."
Having said that, I am talking with our senior Jumpmaster, this topic piqued his interest as well, in order to be completely thorough and to clarify whether this has ever been official policy.
I also want to add that I have in the past served as both a DZSO for US Navy airborne operations and a Safety swimmer for US Army Airborne ops overseas, the type of sheath for a knife was never specified.
Navman, thank you for the informative and current info on the matter. And thank you very much for your service!