Hi AnalogJ,
The breathability of a fabric will depend on the thickness of the fibers and the tightness of the weave more than the type of material. Where they would be most different would be in the ability of each fiber to absorb and wick moisture. Natural fibers can absorb moisture inside the fiber itself while synthetic fibers don’t. This is the main reason that natural fibers can be better at regulating temperature than synthetic fibers but regardless of which type of fabric is used to wrap the coils the innerspring system in the mattress will still have plenty of space outside of the coils and would be a breathable component. The innerspring will also have less effect on the ventilation and temperature regulation of the mattress than the foam layers that are on top of it and the fabric and materials used in the quilting and ticking of the mattress (along with the type of mattress protector you use and the type of sheets and bedding). There is more about the variables that can affect the sleeping temperature of a mattress in post #2 here but I wouldn’t be concerned about the effect of any innerspring on the sleeping temperature of a mattress because any innerspring will allow for plenty of airflow.
Phoenix