No amplification with LMV358 compared to LM358 in crude current sensing
Your problem is that the offset voltage is in the wrong direction (50:50 chance) so your signal is swamped by the offset. If it was the other way you'd see a positive offset in the output voltage. 

The LMV358 has a maximum offset voltage of 7mV, and it could be positive or negative. That's similar to the LM358 but you happened to get one that is negative so you won't get any output until the input exceeds the offset voltage. 

I suggest you use a more precise op-amp that has a lower maximum Vos. Chopper ("Zero drift") types are often used in current sensing since (for obvious reasons) you want to keep the sense voltage as low as possible while still getting acceptable accuracy. Eg. This series of op-amps from ONSemi which is literally hundreds of times better. Note the maximum 5.5V supply voltage.