How to measure small currents using current to voltage converter?
Get a copy of LTSpice and simulate the circuit, you will learn a lot!

Please also let us know what where your reference point is.

Please label you resistors for easy reference. 

What you have is an inverting Opamp circuit.

In the second case, the 20ohm and 240 ohm resistors make a voltage divider.

The Opamp will never drive to the negative rail (even so called rail to rail versions have a small voltage drop). 

Putting a probe on an opamp input is NEVER a good idea, the impedances are just too high. 

The output of the opamp should go to near the negative supply(perhaps even a volt or so depending on the type of opamp), or about -5V5 to the centre 'virtual earth' formed by the 51K resistors.

This is probably where your discrepancy lies.

Your voltmeter impedance might also be skewing with the 51K resistors. 

Rather make a low impedance virtual earth using another opamp as a follower with the feedback resistor at zero ohms, and put a decoupling cap on that to smooth it.

Then retest.

When measuring currents, you would normally have largish resistors in series with the inputs with a shunt (&lt; 1 Ohm) between the probes.

These circuits are all well described in the literature from any manufacturer.