Applying a voltage at the inputs while the power supply is off violates the absolute maximum ratings. (A current will flow through the input ESD protection diodes into V+.)
The purpose of the shutdown function is to allow the opamp to stay connected to…
Thanks Kento for posting your question and Clemens for your response.
Unfortunately, LMV331 does not have the same pinout as TL331. A part that would be same layout for you to consider instead is a part that will be releasing in August (TL331LV-SOT23…