I store fabric in my mother's dresser which is over 70 yeasrs old and have had no problem at all. I jst put the folded fabric in and that is it, no special paper or treatment. Have been doing this for about ten years.

I have wooden shelves and my solution was to lay batting cut to the right size on the shelves to create a barrier between the fabric and wood. You could do this in the drawers. Wood will discolor fabric.
peace

Wood might ooze residual oils and ruin the fabric, so I would say to definitely line the drawer.

several folded layers of an old sheet would work well. Or several layers of acid free tissue.

I would definitely want to put something down to protect the fabric in the quilts from the raw wood. I normally hang my UFOs on a hanger, thus they don't need to be press them again when I am ready to quilt them.

I line dresser drawers with old pillowcases and have had no problems with the fabric in the drawers. This is the one time that plastic would be superior!

Don't forget that drawers, unlike shelves, have sides that also make contact with the fabric. A simple layer of something on the bottom of the drawer will not prevent acid stains from the sides.

Urethane or paint on all the interior drawer surfaces is the best solution. Contact paper is a good alternative and it does not act like a plastic bag since it does not totally enclose the fabric and restrict air flow.

I hope those are not wire hangers. They have been known to rust. So use plastic hangers or leave on the cardboard piece that used to come back from the dry-cleaners with pants. Or make your own wire cover.

I know you should not put quilts into plastic bags, but I wouldn't think laying them on plastic would not do any harm. They need air, and if the plastic is not sealed, you should be OK with the contact paper. What do others think about this?

Agree with the others: generously line the drawers with acid free tissue paper or a clean white towel and you should be okay.