Do Random Forest overfit?
Every ML algorithm with high complexity can overfit. However, the OP is asking whether an RF will not overfit when increasing the number of trees in the forest.

In general, ensemble methods reduces the prediction variance to almost nothing, improving the accuracy of the ensemble. If we define the variance of the expected generalization error of an individual randomized model as:



From here, the variance of the expected generalization error of an ensemble corresponds to:



where p(x) is the Pearson’s correlation coefficient between the predictions of two randomized models trained on the same data from two independent seeds. If we increase the number of DT's in the RF, larger M, the variance of the ensemble decreases when ρ(x)&lt;1. Therefore, the variance of an ensemble is strictly smaller than the variance of an individual model. 

In a nutshell, increasing the number of individual randomized models in an ensemble will never increase the generalization error.