Neural network based receivers have recently demonstrated superior system-level performance compared to traditional receivers. However, their practicality is limited by high memory and power requirements, as separate weight sets must be stored for each code rate. To address this challenge, we propose LOREN, a Low Rank-Based Code-Rate Adaptation Neural Receiver that achieves adaptability with minimal overhead. LOREN integrates lightweight low rank adaptation adapters (LOREN adapters) into convolutional layers, freezing a shared base network while training only small adapters per code rate. An end-to-end training framework over 3GPP CDL channels ensures robustness across realistic wireless environments. LOREN achieves comparable or superior performance relative to fully retrained base neural receivers. The hardware implementation of LOREN in 22nm technology shows more than 65% savings in silicon area and up to 15% power reduction when supporting three code rates.