There is not a great deal of research that indicates which plants are injured, but plants do have varying tolerances to softened water. In many cases, water from a mechanical softener does have a harmful affect on plant growth.
Hard water contains large amounts of calcium and magnesium.
These minerals are the ones that reduce the effectiveness of soap. The minerals combine with the soap to form the ring that often forms in the tub or washbowl. The original calcium and magnesium in the water are plant nutrients. The sodium however is only needed in very small amounts and over long periods of time, can become toxic to the plant. The sodium also has a tendency to damage the soil quality around the root system.