If it really is greasy finger marks, stop eating fish and chips in the workshop or get a water stone instead of an oil stone!
However, I used to have a collegue who had problems with acidic sweat. He used to leave grey/blak finger marks on the wood, especially if it had a high tannin content. It was so bad that his tools had rusty finger marks on them.
I also have a student who is a farrier, he has had problems with his timber getting darker as the project progressed, caused by ingrained dirt in his hands (much of it ferrous) getting onto the timber.
The only solutions I can suggest is a thorough hand was before starting work and application of a barrier cream.