Time-varying distortions in communication systems can significantly degrade the performance of soft-decision forward error correction. This paper presents a burst-aware (BA) low-density parity-check (LDPC) decoding scheme for channels affected by bursty phase noise. By applying differential coding to a Wiener process with time-varying innovation variance, bursty differential phase noise is obtained. Simulation results demonstrate that, compared to conventional decoding, the BA scheme achieves gains in the signal-to-noise ratio of up to $0.7$~dB at a bit error rate (BER) of $4\cdot10^{-3}$ and more than $1$~dB at a packet error rate (PER) of $1\cdot10^{-2}$. Furthermore, by iterating between channel estimation and \ac{ldpc} decoding, forming the proposed iterative burst-aware (IBA) decoding scheme, the gains increase to $1.4$~dB and more than $3$~dB, respectively. More importantly, the IBA scheme significantly improves robustness to bursty phase noise. Compared with the conventional scheme, the IBA scheme can reduce both \ac{ber} and \ac{per} by up to two orders of magnitude under severe bursty phase noise.