It can, it depends on how much stain you use. If you use too much the wood will soak it up and it will change the sounding capabilities of that striker. Certain strikers sound a different way because of the density of the wood, oak is denser then poplar, hickory is denser then oak for instance. If too much stain fills in the "open" areas the striker may lose its tone.
Real question is why worry about it? If turkeys are that close, then you shouldn't be calling anyway, or should have switched to a mouth call.
I honestly would worry very little about them seeing the striker, much bigger fish to fry.