This paper investigates the asymptotic performance of bit-interleaved coded modulation (BICM) with iterative multiuser detection and decoding in uplink sparse code multiple access (SCMA) systems. The extrinsic information transfer (EXIT) characteristics analysis of the joint multiuser detector for SCMA is provided, and shows that the average detection reliability for multiple users converges to the single-user case, if ideal feedback from the decoder is available to the detector. We develop a tight analytical bound on the convolutionally encoded bit-error rate (BER) for independent Rayleigh fadings, based on the single-user bound with arbitrary multidimensional constellations. Moreover, we analyze the achievable coding and diversity gains of the SCMA-BICM system with iterative receiver. Simulations are carried out to verify the effectiveness of the analysis.