Anything made with wood will have some acids in it unless it has been treated to be "acid-free" or "archival" quality. This includes paper and includes tissue paper; paper is made from wood.
Since the dresser is 28 years old, the wood has had a chance to dry out so it won't leak acid as fast as newer wood. However, you still would not want to store fabrics in direct contact with the wood.
My suggestion would be to get a can of polyurethane from the hardware store and give the insides of the dresser drawers a couple of coats. Alternatively, lining the drawers with muslin would be sufficient. Or, store all the fabrics and quilt tops inside of old pillowcases.