Applied 370degC at the input of an ADC in a Microcontroller
What would have happened at the ADC input pin due to this mistake?


It's likely that an ESD event occurred - maybe the circuit was unearthed and had attracted some charge (not a problem by itself but you have to know how to handle these events). The worst thing you can do is earth one single pin and, by the sounds of it, that is what happened when the soldering iron tip touched the wire connected to the pin.


  And what would the same scenario if those pins were output (open drain
  or push pull)?


More than likely it would still damage the chip.

Even an unearthed soldering iron tip can be a huge problem - capacitive coupling to the tip from the AC supply can push a few mA into the chip's pin and cause damage. See this from the data sheet for the part you used: -