As with many domains of computer vision recently, deep learning is not needed to perform the task but does outperform non-DL based methods.
There is lots of prior work that uses other techniques, for example SVMs, to perform facial expression recognition. However if we look at more recent work like the FERA 2017 competition, for example, all of the top performing models are deep learning based models.
Deep learning models are often more difficult to train or require longer development periods than more simple machine learning models, so whether or not you need deep learning for your task is a very difficult question to answer in the general sense and will depend on what your constraints in terms of development time, accuracy, and execution time are.
solved Is Deep Learning really needed for Facial expression recognition? Or Is this just like proposing saw for cutting cake?