What Can Go Wrong?
Fingerprints!  The balance measures the net downward force.  For example, if the weight of the sample is 1.0 mg and the beaker weighs 100 grams, then the accuracy of the balance must be 1 part in 100,000. 
Oily fingerprints do have a measurable weight on this level. 
The most accurate procedures therefore use tongs or lab tissues to handle objects that must be accurately weighed.