Online boosting (training on the fly) for face detection
If you are asking about online learning using OpenCV's Cascade Classifier, I'm pretty sure it is not possible. The reason is that Cascade Classifier consists of number of simpler classifiers which are decision trees in case of OpenCV's face detection system. I don't know exactly, what decision tree algorithm is used, but as far as I know there are no algorithms supporting online learning (such an algorithm would incorporate tree restructuring which is quite inefficient). So, if the base "bricks" do not support online learning, the whole "building" will not support it either. 

On other hand, if you ask whether it is possible in general, then the answer is "yes". All you need is to find algorithm that supports online learning (such as artificial neural networks) and adapt the whole face recognition system to support it. You may still use cascade structure and haar-like (or even newer local binary pattern) features, but base classifiers should be replaced. 

However, in practise I would say "don't do it". Or at least search for simpler options. It is quite a lot of work, both - research and development.