You shouldn’t finish the inside of a cedar chest. Leave it totally unfinished. The reasons are, first, you don’t want to seal in the cedar aroma that protects against moths, and second, the cedar fumes will destroy (soften to the point of stickiness) most finishes.
If the outside of the chest is also cedar, you may have problems getting oil, varnish or polyurethane to dry well. If you want to use one of these finishes, apply a sealer coat of shellac first. For example, use SealCoat, available from paint stores.
If the outside is any wood but cedar, you can use any finish. So choose one you’re familiar with or that will give you the look and durability you want.
