Recent advancements in polar coding have achieved practical performance competitive with other capacity-achieving codes such as low-density parity-check codes. In this paper, we propose a new family called irregular polar codes, where polarlization units are irregularly inactivated to achieve additional degrees of freedom for code design. We first discuss the code construction for irregular polar-coded modulation by taking non-uniform bit-reliability into consideration. We then apply the proposed polar codes to wireless massive multiple-input multipleoutput (MIMO) communication channels. Simulation results show that the irregular polar codes can significantly reduce encoding/decoding complexity up to 50% while also yielding a marginal improvement in error rate performance.