Stochastic modeling of gene expression is a classic problem in theoretical biophysics, and the burst approximation is widely used to simplify gene expression models formulated via the chemical master equation. However, the approximation error has been investigated only for the simplest case. This article proposes and analyzes a general stochastic gene expression model with an arbitrary number of gene states, and quantifies the error introduced by the burst approximation. Using the standard binomial moment method, we derive recurrence relations for binomial moments in steady state. We develop an algorithm to numerically compute binomial moments in a hierarchical manner. In particular, explicit expressions for low-order moments are presented. Compared with surrogate models under the burst approximation, we conclude that the first-order moment of protein counts is preserved, whereas discrepancies generally arise in higher-order moments. By estimating the difference between two second-order moments using functional analysis, we evaluate the validity of the burst approximation.