Reconfigurable intelligent surface (RIS) as a promising technology has been proposed to change weak communication environments. However, most of the current resource allocation (RA) schemes have focused on RIS-assisted homogeneous networks, and there is still no open works about RA schemes of RIS-assisted heterogeneous networks (HetNets). In this paper, we design an RA scheme for a RIS-assisted HetNet with non-orthogonal multiple access to improve spectrum efficiency and transmission rates. In particular, we jointly optimize the transmit power of the small-cell base station and the phase-shift matrix of the RIS to maximize the sum rates of all small-cell users, subject to the unit modulus constraint, the minimum signal-to-interference-plus-noise ratio constraint, and the cross-tier interference constraint for protecting communication quality of microcell users. An efficient suboptimal RA scheme is proposed based on the alternating iteration approach, and successive convex approximation and logarithmic transformation approach. Simulation results verify the effectiveness of the proposed scheme in terms of data rates.