We study the distributed nonconvex resource allocation with Limited Communication Data Rate (LCDR) over a communication network. Each node in the network has its own private cost function and determines the optimal resource allocation through interactions solely with its neighboring nodes. The nodes need to cooperatively minimize the total cost function to achieve the optimal resource allocation under the constraint of constant total resources. First, we consider exact communication. By Lagrange dual method, we propose a successive convex approximation-based distributed dual gradient tracking algorithm to solve the distributed nonconvex resource allocation problems. Then, we consider the case of digital communication among nodes based on LCDR. The information transmission among nodes is based on the Dynamic Encoding and Decoding (DED) with finite-level uniform quantization. We propose a successive convex approximation-based distributed dual gradient tracking algorithm with LCDR and conduct numerical simulations. The numerical results show that the algorithm converges based on merely one-bit quantizers when appropriate step sizes and scaling functions are chosen.