What happens to the left over unpicked data in Random Forest
I suppose it is possible that not all samples are selected during training, depending on the parameters you specify (or that are available in the implementation).

Looking at Scikit-Learn's RandomForestClassifier documentation, we can see that there is a bootstrap argument that can be set to False to ensure all data points are used to fit each of the trees. Otherwise, say you pick some arguments to all be 1 (num_estimators, max_depth, min_samples), then not much data would be used at all! Looking through the source code, there doens't seem to be a check that all data was used.

Another classifier, ExtraTrees (Extremely Randomised Trees) is generally designed to used all samples to train each estimator. However the SciKit learn implementation does allow you to disable that and use random bootstrapping, as is default with the other random forest algorithms.

So to answer your question; it seems the unused samples are simply left out!