The design of feedback channels in frequency division duplex (FDD) systems is a major challenge because of the limited available feedback bits. We consider non-orthogonal multiple access (NOMA) systems that incorporate reconfigurable intelligent surfaces (RISs). In limited feedback RIS-aided NOMA systems, the RIS-aided channel and the direct channel gains should be quantized and fed back to the transmitter. This paper investigates the rate loss of the overall RIS-aided NOMA systems suffering from quantization errors. We first consider random vector quantization for the overall RIS-aided channel and identical uniform quantizers for the direct channel gains. We then obtain an upper bound for the rate loss, due to the quantization error, as a function of the number of feedback bits and the size of RIS. Our numerical results indicate the sum rate performance of the limited feedback system approaches that of the system with full CSI as the number of feedback bits increases.