Linings are usually made of some form of rayon (cupro bemberg is the Italian term that's often used), or occasionally silk. (I've only seen that in the Escorial suits by Brioni, though.) How are the linings different from OTR suits you own? Colour? Texture? Hand? As for the broader question about custom suits, custom is just that. You tell the tailor what you want. S/he makes it happen. If you want the lining made out of petrified wood, that's what you'll get. Won't be very comfortable, though
That said, often really good custom suitmakers will encourage a customer to specify an unlined suit. These are actually more expensive to produce, because the seams that would be hidden by the lining must be picture-perfect instead of simply functional.