I assume you are asking about the Linseed oil you put on your stock? BTW, you didn't say if you were using RAW linseed oil or BOILED linseed oil. There is a difference. With Raw linseed oil, it takes about 6 months for the stuff to dry!
To remove it, dampen a rag with alcohol- rubbing, isopropyl, or denatured alcohol-- they all work. and wipe off the stock. You can use paper toweling for this, but the alcohol evaporates off the paper faster than off a cloth rag.
The alcohol will also soften the original finish that is on the stock, if you leave it on too long, or put too much alcohol on the stock. That is why I suggest you put the alcohol on a rag, instead of on the stock. Because the finish will be soften, when its NO LONGER "STICKY", let the stock sit out, preferably in the sunlight, to dry thoroughly.
Then use a wax furniture polish, or any wax you have( I have used car wax) to coat the stock. The waxes also have DRIERS in them, so they will also help pull moisture out of the finish and help it dry and harden. Use wax to seal the finish after you clean the gun. I only use oil finishes on a stock if it has several scratches on it from use, and I want to cover up some of those "war wounds". Some scratches are deep enough that you will need to stain the wood also, before refinishing. Tru-oil from Birchwood Casey seems to be the easiest oil finish I have used to patch scratches. It blends in with the rest of the finish very easily, and you won't see the Patch if you rub the oil into the scratch and surrounding wood with the butt of your hand.
A quick wash with alcohol, and then soap and water gets the finishes off your hands.