I'm from CA and always have heard them called "dresser Scarves"....My Grandmother made a lot so we always covered whatever with one of them. I used to turn a lot of them into pillow covers, then I put them on the bed to show the work off. I have no idea what one would do with them today. But if it's for you as a memory, I would get creative and if it is thin, yes, I would back it with some muslin...good quality.......

Speaking of old embroidered dresser scarves, I used to collect vintage items like this and made a valance for my guest room in a crazy quilt fashion for the double windows. I like the way it turned out.

I am no expert on restoration of precious or antique items. I did have to mend a quilt for a friend recently. A block was seriously damaged and worn. The fabric that had disintegrated was a very soft almost tissue like fabric. I back it with a very light weight interfacing just to give it more body while I stitched the heavier cotton fabrics. I thik so much depends upon how you plan on displaying this, what other fabrics you are going to use etc. Will the light weight linen sag? Can it "hold it's own" with a batik? A couple of years ago I made a simple white dll using an old pillowcase that my Grandmother had embroidered. The embroidery was the doll's skirt and the crochet lace trim was the doll's bonnet. It was the most sentimental precious thing I've ever made!

I would back it with a very light interfacing. The fabric you add to it could possibly pull or tear your old fabric. If you were going to add embroidery, use a stabilizer.