Why do sungold kiwis never sting the tongue, while green kiwis can?
The "sungold" kiwifruits are certainly a specific cultivar of kiwi. I don't know enough to tell you if all the green kiwis you are buying are from the same cultivar or not. 

Also, notice that cultivars of the same plant are rarely "pure" enough to always have the exact same chemical characteristics, and sometimes even a pure cultivar will express different levels of a given chemical in different growing conditions. For an example of the first, I have observed peppers from the same seed stock, sown in the same bed, to have different heat, sometimes it can even happen that two different fruits of the same plant have different heat. For the second, a watermelon's sugar content will be very dependent on the amount of sun received during growth. 

Kiwifruit contains an enzyme which attacks protein, and that's what produces the stinging - your mouth's proteins are being injured. The levels in kiwi are much lower than in fruits like pineapple though, so frequently the effect is not present at noticeable levels. I don't know of a very special reason why they are never noticeable in the sungold variety, it's possible that it was bred for this purpose (along with the higher sugar content) or just that it is an evolutionary spandrel.