I've got a quiver lined with fleece and I hate it. It's a PITA trying to put arrows back into it because the points really do snag up in the fleece. I like my quivers unlined, but maybe with some fleece or fur around the top edge to help keep the arrows from rattling.
You can keep the points from rattling around by dropping a couple of handfuls of cedar shavings into the bottom of the quiver. They sell bags of cedar shavings in the pet store. I guess the unwashed,uninitiated and uncivilized non-archers use it to putin the bottom oftheir hamster cages.

Never overlook the pet store when you're shopping for archery supplies.