There is water coming out of the floor where the copper radiant heating pipe exits the floor. Is there any way to locate the leak before tearing up the entire tiled floor?

You can hire a leak detection company, American Leak Detection is one I am aware of, at least in the Atlanta Area. Companies use ultrasonic and radioactive isotopes to try and find the leak. Your issue is why I am not a fan of in floor heating, you may have more than 1 hole in the copper line. Now they almost always use PEX.