I am getting ready to finish my basement. I can use Tapcons or a Ramset to attach the bottom plates of the walls to the concrete slab with drain tile underneath. Will these holes from the screws or nails allow radon in?
If you have radon, it's already likely getting in--concrete is permeable. So adding a small nail hole isn't going to change anything. Also note that tapcons don't actually penetrate all the way through the slab.
If you have radon, you need to get a radon remediation system installed. This will put a hole in your slab so the radon has a place to go (and then it's vented out the roof).