Bursty mixed noise is typically characterized by the combination of white Gaussian noise (WGN) and non-Gaussian impulsive noise (IN) with memory. The performance of communication systems can experience significant degradation when conventional signal detection algorithms designed for Gaussian noise are applied in the presence of bursty mixed noise. To handle this challenge, we investigate demodulation techniques and conduct a theoretical performance analysis under bursty mixed noise. Specifically, we utilize the statistical model of channel noise to derive demodulation methods for both linear and nonlinear modulation schemes. We further analyze the theoretical error probability for M-ary PSK/QAM and provide closed-form expressions. For MSK/CPM modulations, we derive asymptotic lower and upper bounds for the error probability, accompanied by analytical expressions. Simulation results demonstrate that the proposed demodulation methods outperform baseline techniques by more than 1.9 dB when the error probability approaches 10-3. Additionally, the theoretical analysis closely aligns with the simulation results.