Yes even peeled potatoes will go bad even when they are kept directly in the refrigerator.
To keep peeled potatoes fresh in the refrigerator the peeled potatoes should be submerged in water in a bowl and they will last longer.
Take the peeled potatoes and then submerge them in a bowl of water, cover and refrigerate them.
When you store the potatoes submerged in water the potatoes should last a week or longer but if you just put the peeled potatoes directly into the refrigerator without the water then they will turn dark overnight.
The potatoes will still be okay and safe to eat after one night and being dark but keeping in the covered bowl of water will help them last longer in the fridge.