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Dynamic graph neural network (DGNN) is becoming increasingly popular because of its widespread use in capturing dynamic features in the real world. A variety of dynamic graph neural networks designed from algorithmic perspectives have succeeded in incorporating temporal information into graph processing. Despite the promising algorithmic performance, deploying DGNNs on hardware presents additional challenges due to the model complexity, diversity, and the nature of the time dependency. Meanwhile, the differences between DGNNs and static graph neural networks make hardware-related optimizations for static graph neural networks unsuitable for DGNNs. In this paper, we select eight prevailing DGNNs with different characteristics and profile them on both CPU and GPU. The profiling results are summarized and analyzed, providing in-depth insights into the bottlenecks of DGNNs on hardware and identifying potential optimization opportunities for future DGNN acceleration. Followed by a comprehensive survey, we provide a detailed analysis of DGNN performance bottlenecks on hardware, including temporal data dependency, workload imbalance, data movement, and GPU warm-up. We suggest several optimizations from both software and hardware perspectives. This paper is the first to provide an in-depth analysis of the hardware performance of DGNN Code is available at https://github.com/sharc-lab/DGNN_analysis.