OPA2376: What is happened on used amp side of OPA2376 when both input pins of unused amp are connected to GND?

That is actually not a good idea, 

Running the unused op amp open loop with grounded inputs will still put the output at one rail or the other, possibly drawing more current than necessary - this is due to the offset voltage and high DC Aol. 

Normally, you would run the unused side as a voltage follower (output connected to the inverting input) and the V+ input set to keep the Vo in range (probably not ground). 

Kazuya-san,

An unused op amp of multi-channel package version should be connected in the way so its input and output operate within its specified linear range - see below.

Connecting both inputs to ground in a single supply application will result in the op amp output collapsing close to one of its rails - which rails depends on the polarity of the Vos. In the bipolar transistor output stage saturating the output close to rail would lead to the sudden drop of the transistor beta resulting in the increase in the output stage current BUT since OPA376 uses only CMOS transistors, there should be no significant change in the IQ current.

 Hello Marek,

 Thank you very much for your reply.

 Do you mean the unused opamp inputs can be connected to GND in OPA2376 case because IQ is not increased?

 Is there any effect on the performance of the used opamp side?

 Could you give me your comment?

 Thank you again and best regards,

 Kazuya.

The unused op amp should NOT have its inputs connected to negative rail (ground) but doing so with CMOS op amp should not cause an increase in the quiescent current or negatively effect other channel of OPA2376.  In other words, the customer should not do it because this forces the output to operate in a non-linear region but this will not cause a major issue if they do.