I'm getting a free binder from Amos (sort of)

I would Scott to send you a new one!

I've had good luck removing grease and oily marks with "Goo Gone" on a rag. Not to be confused with another product called "Goof Off" which I've used for more extreme problems but it's basically a paint remover, so maybe don't try that on your binder. A small amount of Goo Gone on a rag is not likely to damage anything and may work to remove whatever is staining that binder, but test a small amount just to be sure it does no harm.

And I just remembered that Dawn dish detergent is famous for being a grease and oil remover while being very safe and essentially harmless (even used on birds fouled by oil spills). Maybe try a few drops of that on a wet rag first?

I really don't think that binder will take to paint of any kind which is likely to just make it worse -- and perhaps a mess. And if that's a Scott binder, then it's covered with the product they've used for years which is a kind of "vinyl-like" heavy paper or fabric they call "fabrikoid". It's somewhat washable if you don't go crazy with the amount of water. I often clean Scott binders with some hot water on a rag and maybe a drop or two of Dawn detergent. Never had a problem and it works for me. Alcohol and other such products work as paint removers, so nix them. Light oils like Goo-Gone may work. I wonder if a small amount of a light oil like WD-40 would work -- but I'd be a bit scared to try it on a binder like that.