Can the skin of lychee be used for anything?
The rind is inedible: https://en.wikipedia.org/wiki/Lychee

I've had a lot to do with Asian culture, both in and out of the realm of the culinary arts, and have never seen or heard of lychee rind used for anything. There are some fruits like lime, lemon, grapefruit, etc., wherein the rind can be used for other purposes, or even eaten. However, in this case, I think the most you could do to make use of it is to compost it and use it as nourishment for a garden. -- Which would certainly help feed your plants. :)

Another idea that is that sometimes things are put on the plate purely for decoration. The peel could be cut into decorative shapes and put on the plate with the peeled lychees, if you have guests over. You just need to make sure they know the peel is not edible. I've never tried this before, so I cannot attest to how well they work for this purpose, but it's an idea more relevant to cooking/presentation.