A setup involving zero-delay sequential transmission of a vector Markov source over a burst erasure channel is studied. A sequence of source vectors is compressed in a causal fashion at the encoder, and the resulting output is transmitted over a burst erasure channel. The destination is required to reconstruct each source vector with zero-delay, but those source sequences that are observed either during the burst erasure, or in the interval of length $W$ following the burst erasure need not be reconstructed. The minimum achievable compression rate is called the rate-recovery function. We assume that each source vector is sampled i.i.d. across the spatial dimension and from a stationary, first-order Markov process across the temporal dimension. For discrete sources the case of lossless recovery is considered, and upper and lower bounds on the rate-recovery function are established. Both these bounds can be expressed as the rate for predictive coding, plus a term that decreases at least inversely with the recovery window length $W$. For Gauss-Markov sources and a quadratic distortion measure, upper and lower bounds on the minimum rate are established when $W=0$. These bounds are shown to coincide in the high resolution limit. Finally another setup involving i.i.d. Gaussian sources is studied and the rate-recovery function is completely characterized in this case.