Measuring voltage across 1 ohm resistor
If you were using an analog multimeter with, say, 20,000\$\Omega\$/V on a 0-150mV F.S. range, the meter looks like an approximately 3K resistor. 

Thus the voltage across the 1\$\Omega\$ resistor will be a bit less since it is shunted by about 3K (3002 ohms if you count the leads) and the voltage making it to the 3000 ohm meter movement is a bit less as well (3000/3002 of the voltage across the 1\$\Omega\$ resistor) because 1/3002 of that voltage is dropped across each lead. 







simulate this circuit &ndash; Schematic created using CircuitLab

In total then, in this example, the voltage read at the meter is lower than ideal by about 0.1%, which is going to be totally insignificant compared to the tolerance of the resistors and the accuracy of the meter. 

A typical digital multimeter will have  an input resistance of something like 10M\$\Omega\$ on the 199mV scale so the effect will be several thousand times less, so even with a very accurate resistor and meter it's negligible. Not to mention that the leads are probably closer to 0.2\$\Omega\$ than 1\$\Omega\$.