When pin IN- withstands 12V, how much current does it sink?
At this time, the other pin voltages are 0V
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It could sink many mA. As Kai pointed out, the Abs Max is 10mA. Beyond that, damage could occur.
The inputs have diode clamps to the supplies:
If you hold V+ at 0V, then the upper ESD diode will conduct into the supply.
Depending on the amount of input current available, you could back-feed the V+ supply with the input voltage and power your circuit.
You will need to limit the current - usually with a series resistor on the input. We recommend at least 1k per volt to limit current to 1mA, so you should have a minimum of 12k series resistance. The more the better for less current. This resistance can be part of any input dividers.
If this is occurring during normal operation, then external Shottky diode clamps to the supplies are recommended to take the strain off the internal ESD structures.
0V supply is not within the specified operating conditions, so operation (or behavior) is not warrantied.