Baseline correction is a critical step for eliminating the interference of baseline drift in spectroscopic analysis. The recently proposed sparse Bayesian learning (SBL)-based method can significantly improve the baseline correction performance. However, it has at least two disadvantages: (i) it works poorly for large-scale datasets and (ii) it completely ignores the burst-sparsity structure of the sparse representation of the pure spectrum. In this paper, we present a new fast burst-sparsity learning method for baseline correction to overcome these shortcomings. The first novelty of the proposed method is to jointly adopt a down-sampling strategy and construct a multiple measurements block-sparse recovery problem with the down-sampling sequences. The down-sampling strategy can significantly reduce the dimension of the spectrum; while jointly exploiting the block sparsity among the down-sampling sequences avoids losing the information contained in the original spectrum. The second novelty of the proposed method is introducing the pattern-coupled prior into the SBL framework to characterize the inherent burst-sparsity in the sparse representation of spectrum. As illustrated in the paper, burst-sparsity commonly occurs in peak zones with more denser nonzero coefficients. Properly utilizing such burst-sparsity can further enhance the baseline correction performance. Results on both simulated and real datasets (such as FT-IR, Raman spectrum, and chromatography) verify the substantial improvement, in terms of estimation accuracy and computational complexity.