The support lining usually consists of burlap, with jute webbing straps underneath.
The support linings hold the padding in place. It is important that they are in good repair if you want the sofa to stand up to much use. To replace the support linings you generally need to remove the padding, taking pictures and keeping a keen eye as to how the padding is attached. After the padding is off, examine how the old support linings and webbing was attached and (providing it was professionally done previously) put the new burlap and webbing on the same way. Then, either put the existing padding back on, or replace with new padding, replacing it the same as it came off.
Most modern sofas will have polyfoam in the cushion. Even the cushions had springs or something else in them, you can still put new foam cushions in them. If you will be putting a zipper in the seat cushions (which is recommended) you can always replace the foam in the cushions at a latter date). When replacing the foam, use a high quality foam of at least 2.5 lb to 3 lb. which will give you many years of service.