What should I do with the unused Op-amp inputs?
It depends a bit on the op-amp. If it's a bipolar op-amp with high input bias current in a non-precision circuit (referring to the other op-amp) it may be safe to leave the inputs open. Think LM324 in some non-critical circuit. 

One common practice is to connect the op-amp as a voltage follower and tie the non-inverting input to some voltage within the input and output common-mode ranges. Since the unused op-amp is operating normally, all the datasheet guarantees apply, so this could be considered a best practice approach. 

You can also tie the inputs to potentials such that the output (left open) is railed high or low (there may be some small power consumption advantage). 

I don't suggest connecting the inputs to the same potential since the output of some units might drift around and slightly affect the other op-amp. Similarly, leaving inputs open on an amplifier with low input bias current (those with bias current cancellation or JFET/CMOS input op-amps) can cause pickup that will show up in power consumption and can show up as crosstalk in the amplifier that is being used. 

Edit: As the actual amplifier OPA2614 appears to be a non-unity-gain stable very high frequency type, it would likely be best to either make a gain-of-five amplifier or apply a small DC voltage of a few hundred mV on the inputs and leave the output railed.