Bit-interleaved coded modulation with iterative decoding (BICM-ID) is a key technology enabling communication systems to realize high-throughput and high-reliability. In contrast to previous studies focused exclusively on uniformly distributed sources, this work investigates irregular mapping (IM)-based BICM-ID systems for sources with non-uniform distributions. First, a density evolution (DE) algorithm based on Gaussian weighted mixture approximation (GWMA) is put forward to estimate the convergence performance of the IM-BICM-ID systems with non-uniform sources, where the probability density function of the log-likelihood ratio messages output from the bit-level channels in this system is neither Gaussian nor symmetric. Then, by employing the GWMA-DE algorithm and the achievable system rates analysis, a design strategy for optimal IM labelings is established, which utilizes the inherent redundancy in non-uniform sources to achieve shaping gains. Furthermore, channel codes tailored to the proposed IM labelings are designed to further improve the bit error rate performance. Both analytical results and simulations indicate that the proposed labelings and codes outperform existing counterparts in IM-BICM-ID systems with non-uniform sources.