Utilizing edge and cloud computing to empower the profitability of manufacturing is drastically increasing in modern industries. As a result of that, several challenges have raised over the years that essentially require urgent attention. Among these, coping with different faults in edge and cloud computing and recovering from permanent and temporary faults became prominent issues to be solved. In this paper, we focus on the challenges of applying fault tolerance techniques on edge and cloud computing in the context of manufacturing and we investigate the current state of the proposed approaches by categorizing them into several groups. Moreover, we identify critical gaps in the research domain as open research directions.