The MN is unique (far as comparing to modern firearms, anyway) because the stock encompasses the barrel. The stock contacts the barrel in most I've seen. I would suppose that resting the forend of the rifle on a rest might affect the barrel harmonics and accuracy the same as if you laid the barrel of a new rifle on a rest when shooting.
I have two Mosins- one highly modified in a Boyd's stock with a fully free-floated barrel. The other is original stock, but I still free-floated the barrel by removing stock material from the lower stock and the handguard. Seems to pass the "business card" test. Have not yet had a chance to shoot this one...