Motor shield and Arduino Led not turning on when connected to laptop and Arduino is heating up
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You might have burned one of the onboard voltage regulators. In my case, it was AMS1117 which was burned and my Arduino was behaving similarly. You will notice that the Arduino will work with an external power supply connected to the Vin pin of Arduino and it will not become hot. 

 In short, the small chip which is getting hot is a voltage regulator may be AMS1117 and you have to replace it manually or buy a new Arduino