Rayon is one fabric gradually harmed by "dry" ironing. Silk fabrics should never be "sprinkled" to dampen and iron as the fiber is prone to water spots. Steam ironing (if one's iron will put out steam at such a cool setting), or dry ironing slightly damp fabric is better. The old way would be to take a silk item from the wash and roll it in a towel and allow the moisture to be absorbed until the garment was damp dry enough to iron. The alternative today would be to use a very low rpm final spin. If the item is still too damp see above about the towel.
There is a school of thought that steam ironing benefits all fabrics that can withstand the process (which by nature would exclude many man made fibers unless a cool iron setting is used), because the steam helps protect fibers from the iron's not surface.
Today it is rare outside of commercial irons to find anything with the heat given off by vintage irons or sad irons. Dry ironing is basically drying and smoothing cloth by applying heat and friction. Both can cause fibers to break down if done often enough. Pressing (either by machine or hand) is slightly better as it eliminates friction, but one is still applying heat.
If one irons an area too long past the point when all moisture has evaporated you run the risk of burning or scorch if you will. This often shows up as yellowing on cottons or linens. Indeed the European process of cold mangling/ironing is said to be better for fabrics and keep them white and fresh, that is not using heat.
Have two cold mangles (one manual, one electric) and for some items, mainly linen or cotton fabrics the process works great.
Dry ironing also can damage fabrics if heavy pressure is applied by the operator. This will be especially true on folds/creases such as collars,cuffs and hems. There you'll start to see fraying and other damage caused by abrasion.
Mind you allot of this friction/abrasion damage can be delayed or eliminated by using an iron with a smooth soleplate. In the old days one would apply wax to any iron or ironer to keep the surface slick. Such products are still sold today for commercial laundries, but with the advent of SS,Teflon and ceramic soleplates most home irons do not require waxing.