if for same input values we have different class labels, Should we have to remove those training examples before building any model on it?
No, you should not remove them. 

The problem in your case are not these individual samples. The real problem is that the attributes (variables) that you chose, i.e. A1-A2-A3, are not informative enough to accurately discriminate the samples. This is visible from the fact that the same input space (i.e. X-Y-Z) leads to different classes (A-B). 

If you visualize this in a 3D plot with different colors for each class, you will see that the same datapoint will have two colors (!). This is because two datapoints fall exactly on top of each other and it is impossible to find a linear/nonlinear separation for them, no matter how hard you try.

Therefore, if you wish to increase the classification accuracy, you should pick other attributes/variables and not drop these individual samples; you have to change the way you look at the problem :)