Best Fabric for Sensitive Skin

Yes, any natural fibres. There are so many chemicals in synthetic fibres that can cause irritation to the skin, some of these chemicals can be even cancer inducing. The first step for me was to find out why I had suddenly become allergic to certain things. You could do this with an intolerance test. I now wear only only cotton, linen and in winter 100% wool. I wash in goats milk soap only and I use organic cosmetics. My skin now is as soft as a baby's bottom and I no longer have any problems whatsoever. You can find natural products very cheaply nowadays so don't be put off by thinking they might be expensive.

True.
I have silk pillows cases, and even if my face or hair is greasy at times, they never soak up any moisture.
I'm sure someone will give good advice.

Silk has a low absorbency and can cause buildup on the skin. Bamboo or cotton are my best options and I am hoping to hear opinions from people who wear either of these.

Silk is very gentle.