output offset: dual supply opamp running on single supply
Every time you take the input voltage higher than 0V you are "asking" the op-amp to produce a negative output voltage - this circuit is an inverting precision rectifier - it is trying to make a negative voltage on the output follow the positive voltage on the input. With no negative supply it doesn't work. With a positive bias and no negative supply it doesn't work. 

Try a negative bias on the input or a positive bias on the non-inverting input.

Alternatively swap the diode directions and let a positive rectified output from the op-amp properly follow a negative half cycle on the input.