The use of low-precision analog-to-digital converters (ADCs) is a low-cost and power-efficient solution for a millimeter wave (mmWave) multiple-input multiple-output (MIMO) system operating at sampling rates higher than a few Gsample/sec. This solution, however, can make significant frame-error-rates (FERs) degradation due to inter-subcarrier interference when applying conventional frequency-domain equalization techniques. In this paper, we propose computationally-efficient yet near-optimal soft-output detection methods for the coded mmWave MIMO systems with low-precision ADCs. The underlying idea of the proposed methods is to construct an extremely sparse inter-symbol-interference (ISI) channel model by jointly exploiting the delay-domain sparsity in mmWave channels and a high quantization noise caused by low-precision ADCs. Then we harness this sparse channel model to create a trellis diagram with a reduced number of states or a factor graph with very sparse edge connections. Using the reduced trellis diagram, we present a soft-output detection method that computes the log-likelihood ratios (LLRs) of coded bits by optimally combining the quantized received signals obtained from multiple receive antennas using a forward-and-backward algorithm. To reduce the computational complexity further, we also present a low-complexity detection method using the sparse factor graph to compute the LLRs in an iterative fashion based on a belief propagation algorithm. Simulations results demonstrate that the proposed soft-output detection methods provide significant FER gains compared to the existing frequency-domain equalization techniques in a coded mmWave MIMO system using one- or two-bit ADCs.