Several reasons. One is that our heat sensing nerves don’t sense absolute temperature, they sense how quickly heat is transferring into our out of the area. If the difference in temperature of your hand and the surroundings is large, then there is more heat transfer, and the strong nerve activation is interpreted as burning, even if the temperature isn’t enough to cause a burn.
Another is that when extremities get cold, the blood vessels constrict to reduce heat transfer from the blood. When the extremities warm up and blood flow returns, you feel that same tingling feeling when your arm or leg falls asleep and is starting to “wake up” again. This causes some of the lingering pain after you remove your hand from the warm water.
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