This paper studies stabilization of uncertain systems via data-rate-limited and lossy communication channels. We consider linear systems where the parameters are given as intervals and the exact values are unavailable. Communication between the plant and the controller is restricted in the sense that transmitted signals should be represented in finite bits and may be randomly lost. While most of the existing works assume that the loss process is independent and identically distributed, we model it as a two-state Markov chain, which can deal with more practical situations including bursty dropouts. The central question investigated is how large the data rate, the loss probabilities, and the uncertainty bounds should be to make the system stable. We derive a necessary condition and a sufficient condition for stability. The conditions provide limitations characterized by the product of the eigenvalues of the nominal plant. In particular, for scalar plants, the conditions coincide with each other. Furthermore, we introduce a nonuniform quantizer, whose quantization cells are designed to minimize the effect of the uncertainty on state estimation. The quantizer can reduce the required data rate compared with the conventional uniform one.