Snapshot compressive imaging (SCI) cameras capture high-speed videos by compressing multiple video frames into a measurement frame. However, reconstructing video frames from the compressed measurement frame is challenging. The existing state-of-the-art reconstruction algorithms suffer from low reconstruction quality or heavy time consumption, making them not suitable for real-time applications. In this paper, exploiting the powerful learning ability of deep neural networks (DNN), we propose a novel Tensor Fast Iterative Shrinkage-Thresholding Algorithm Net (Tensor FISTA-Net) as a decoder for SCI video cameras. Tensor FISTA-Net not only learns the sparsest representation of the video frames through convolution layers, but also reduces the reconstruction time significantly through tensor calculations. Experimental results on synthetic datasets show that the proposed Tensor FISTA-Net achieves average PSNR improvement of 1.63∼3.89dB over the state-of-the-art algorithms. Moreover, Tensor FISTA-Net takes less than 2 seconds running time and 12MB memory footprint, making it practical for real-time IoT applications.