One of the main functions of a mattress fabric is to help maintain the shape of the mattress and to help protect the materials in the mattress from exposure to light, ozone, solvents, or other influences that can oxidize or degrade them more quickly.
In some cases a fabric can be the weak link in a mattress and wear out before other layers of a mattress. This however is rarely the case with higher quality fabrics. Fabrics also tend to vary widely in their “elasticity” with knitted fabrics being much more elastic than their woven counterparts… especially those that are tightly woven. Flexible and durable fabrics are an important part of allowing the underlying layers to form a cradle and relieve pressure so it is important that the ticking of your mattress is both durable and flexible. How it is made and how tightly it is attached will also play an important role in altering the qualities of your mattress. Fabric tightness, like quilting, can also be used as a form of zoning however this too is better done in other layers of a mattress.
Fabrics can also be a part of having a more “natural” mattress as there are many available which are “organic” and very high quality. Semi synthetic fabrics like the many types of viscose/rayon materials and natural or organic cotton are popular choices here. Fabrics can also be replaced if they wear out before the layers of a mattress, as there are many sources for zippered mattress replacement covers, with and without quilting layers that can be used to recover your mattress. This can also be done professionally at a reasonable cost as well so all is not lost if you have high quality layers inside your mattress and the cover wears out before they do.