Neural speech codecs have been widely used in audio compression and various downstream tasks. Current mainstream codecs are fixed-frame-rate (FFR), which allocate the same number of tokens to every equal-duration slice. However, speech is inherently non-uniform in temporal information density. As a result, many tokens are wasted on steady-state segments like long vowels and silences. To address this mismatch, we present CodecSlime, a plugin-style method for compressing temporal redundancy through supporting dynamic frame rate (DFR) on neural speech codecs for the first time. Our method is unsupervised and architecture-agnostic, combining two key innovations, ScheDFR and Melt-and-Cool, for adapting inference and training, respectively. When integrated into a typical VQ-GAN codec backbone and operating at 40 Hz DFR ($\approx$ 600 bps), the reconstruction WER of CodecSlime is reduced by up to 32% relative to conventional FFR baselines with the same model architecture and similar bitrates, while other metrics are also competitive. CodecSlime also enables flexible trade-offs between reconstruction quality and bitrate: a single model supports inference at multiple frame rates and consistently outperforms FFR models at the corresponding frame rates. Audio samples are available at https://acadarmeria.github.io/codecslime/.