In this paper, we analyze downlink non-orthogonal multiple access (NOMA) networks with limited feedback. Our goal is to derive appropriate transmission rates for rate adaptation and minimize outage probability of minimum rate for the constant-rate data service, based on distributed channel feedback information from receivers. We propose an efficient quantizer with variable-length encoding that approaches the best performance of the case where perfect channel state information is available everywhere. We prove that in the typical application with two receivers, the losses in the minimum rate and outage probability decay at least exponentially with the minimum feedback rate. We analyze the diversity gain and provide a sufficient condition for the quantizer to achieve the maximum diversity order. For NOMA with $K$ receivers where $K > 2$, we solve the minimum rate maximization problem within an accuracy of $ε$ in time complexity of $O\left(K\log\frac{1}ε\right)$, then, we apply the previously proposed quantizers for $K = 2$ to the case of $K > 2$. Numerical simulations are presented to demonstrate the efficiency of our proposed quantizers and the accuracy of the analytical results.