Current Sensing with Non-Inverting Amplifier
In all probability you are using an op-amp that just cannot work with the inputs close to its most negative power rail (0 volts in your example). Most "regular" op-amps need a couple of volts headroom within the power rails for inputs. This also applies to outputs.

If you are using an LM324 you might get this to work by loading the output with a 1 kohm resistor. It says something to these ends in the data sheet.