Does out-of-bag sampling make Random Forests inherently less robust than other classifiers? All of the popular evaluation metrics (ROC-AUC, Confusion Matrices, etc.) require two lists as parameters: a list of the actual y labels associated with some arbitrary group of training examples (x's), and a parallel list of the predicted labels given to those x's by the model.

To construct such lists, you must separate a testing/validation data set from the training set you feed the model on. However, random forests automatically partition 1/3 of the training set you give it to calculate the out of bag score. I don't believe you can stop the bagging process that causes this, since I think it is vital to how the random forest functions. Because the RF model never sees 1/3 of the training set (due to bagging), do RFs create a less thorough image of the dataset then, say, a neural network would whenever a testing set is reserved for evaluation?
The out-of-bag-error is calculated from the samples that are not used anyway for the particular tree. The original set of records gets bootstrapped. So, a new set is generated that does not contain all samples anyway. The out-of-bag set can then be used to monitor the performance of the model. When the out-of-bag-error goes up (given it has a significant size) it means that the current tree is over-fitting the training sample. So, out-of-bag sampling can be used to prevent over-fitting and therefore makes the model more robust rather than less robust. 

So, in fact all samples are used to train the random forest model. Though each tree only uses a subset of the dataset at a time. Don't confuse the RF model with the individual trees! 

Look here for a more detailed description.