Non-relational database systems (NRDS) such as graph and key-value have gained attention in various trending business and analytical application domains. However, while CPU performance scaling becomes increasingly more difficult, field-programmable gate arrays (FPGA)- accelerated NRDS have not been systematically studied yet. This survey describes and categorizes the inherent differences and non-trivial tradeoffs of relevant NRDS classes (i.e., graph, document, key-value, and wide-column) as well as their commonalities in the context of common design decisions when building such a system with FPGAs. In particular, we highlight accelerator tasks, FPGA placement, accelerator design patterns, and justification for using FPGAs in different system contexts. We close with open research and engineering challenges to outline the future of FPGA-accelerated NRDS.