Will a penny become cleaner in tap water?

Depends what's on it. Tap water would probably be able to remove dirt, but the problem is that when the water hits it, it generally takes less force to go around the dirt than to displace it, so that's what the water does. When the dirt is really loose, then it doesn't take much force to just move it out of the water's path, so it is removed from the coin by the tap-water.

If I remember correctly, the reason the dirt comes off in cleaning fluids is that it reacts with the dirt.