How do I remove the glued on felt from an old antique tool and die tool chest?

I have my grandfather's old wooden tool and die tool chest that I want to use as a jewerly box. The felt lined drawers are dirty, oiley and badly worn. How do I go about removing that old lining which looks like a felt of some kind. Is there some kind of solvent that I can use to help loosen it?
Has anybody done this procedure before?

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