Probably, just to be safe. I've heard of discoloration and even degradation happening because of the acids and stains in wood. I've also heard not to store starched fabrics because it attracts moths and silverfish. I've never had either issue, but I don't store fabric next to wood and I live in an area that doesn't have a lot of silverfish.
You could get some of that adhesive contact paper to line the drawers, then do a layer of acid-free tissue. The reason why I think the contact paper would be a good idea is because tissue is porous and could absorb whatever is in the wood.