The typical input bias current of the LMV721 is 260nA but that is not enough to cause the problem. I just tried it with my simulator and the results don't look right. I suspect there is a fault in the Spice model. I will see if I can spot the problem, but you may want to try another opamp in the meantime or contact National Semiconductor.
Keith.
Added after 24 minutes:
I think the problem with the LMV721 model is that the node order for the subcircuit is mislabeled. This is a National Semiconductor fault, not yours! If I connect positive power to what they say is negative power, ground to what they call the output and take the output from what they label the the positive power it works correctly!
Keith.